Eto'o opened the scoring after just four minutes by planting a header past Tom Heaton from close range following a Leighton Baines cross.
An error from Romelu Lukaku helped Burnley get back into the clash through Danny Ings, but the Belgian hitman atoned for his earlier mistake by restoring the lead on the half-hour mark.
The Toffees looked on course to squeeze by with a narrow win, before Eto'o added a third in superb fashion with five minutes remaining.
Below, you can relive all of the action with Sports Mole's live commentary.
12.30pmGood afternoon ladies and gentlemen and welcome to our live text commentary of Everton's clash with Burnley at Turf Moor. The Clarets are the only side left in the Premier League yet to pick up a victory this season, while Everton have only managed two wins themselves from their opening eight games so this one might be closer than it initially seems.
12.33pmWe will start by looking at Burnley and, actually, they're the only team left in the four tiers of English football still waiting for their first league win of the season. What's more is that, with four goals, no Premier League or Football League club has scored as few as the Clarets this season.
12.36pmIt's not looking good for them, eh? Instead of dressing as the grim reaper for Halloween this Friday you could go dressed as Burnley's statistics because they're no less cheerless than a black cloak and scythe. Let's have a look at their deficiencies in the final third because it is clear that a poor ghoul return - or goal return - is the reason behind their woeful start.
12.39pmLast Saturday, George Boyd joined a very exclusive club of Scott Arfield, Michael Kightly and Ross Wallace by scoring in the 3-1 defeat to West Ham. Yep, it was just their fourth goal of the season. None of their strikers have found the net thus far, which is worrying. The pre-season favourites for relegation aren't doing much to dismiss that label, are they?
12.42pmWith 21 Championship goals last season, Danny Ings played such a pivotal role in helping Burnley reach the promised land but their star striker is yet to make the step up in class to Premier League level. Not only has he failed to find the net, but he has only had one shot in his five top-flight outings. He has, perhaps, missed the partnership with Sam Vokes - sidelined with anterior cruciate ligament injury - after the duo scored 47 times last term, but he must find a way to flourish without the Welshman. Burnley's season depends on it.
12.44pmYep, it is no major surprise to see Sean Dyche's side languishing in 18th place in the Premier League table and, worryingly, you wonder where the goals are going to come from if he is to achieve what is looking like an increasingly difficult task of preserving the Clarets' top-flight status.
12.47pmWe will have a closer look at Everton a little later but let's see what's what with the teams first, shall we? We'll have this very soon, but while we wait, why not check out what the lads at Sports Mole HQ have had to say on this one? See you in a few minutes.
12.55pmRight we'll start with Burnley and I can tell you that Dyche just makes the one change from the 3-1 defeat to West Ham, with Stephen Ward replacing Ben Mee at left-back. David Jones, Michael Kightly, Scott Arfield and George Boyd line up across the midfield, while Lukas Jutkiewicz partners Danny Ings up top as expected. Dyche keeping faith with quite a few of these players and his 4-4-2, then!
12.58pmOkay, on to Everton now and, like Dyche, Roberto Martinez makes one change from his side's 3-0 win at home to Aston Villa last week, with Ross Barkley dropping to the bench for Samuel Eto'o. He partners Lukaku up front, while Steven Naismith, Leon Osman, James McCarthy and Gareth Barry form a decent looking midfield. Seamus Coleman and Leighton Baines line up in the full-back spots, with Antolin Alcaraz and Phil Jagielka operating at the heart of defence in front of goalkeeper Tim Howard. And that's your lot!
1.00pmWe've already discussed Burnley's travails but let's look at Everton and see if we can't churn out one or two positives for the Clarets this afternoon, shall we?
1.03pmOne very positive statistic for Burnley is that the Toffees have lost both games that have come after a Europa League outing, and they played Lille on Thursday. That's a decent enough omen for the hosts. Here's another: Roberto Martinez's side have conceded more goals directly from errors than any other side in the league. They've only won twice themselves so, you know, a maiden win for Dyche and co. can't be ruled out here.
1.06pmTheir participation in the Europa League is testament to how excellent Everton were last season as Romelu Lukaku inspired them to fifth place in the table but, since signing on a permanent deal for £28m, the Belgian hitman is yet to scale those heights this season. Lukaku has only found the net three times so far and, really, Everton need consistency from their record signing if they have aspirations to improve on that fifth-placed finish.
1.09pmIs Champions League qualification too big of an ask for Everton? Probably, but we can't rule it out. They've got some very decent names in their ranks and, as they showed last term, they're more than capable of fighting the big boys for their top-four position. Still, victory here is crucial because they would move within four points of fourth-placed West Ham United.
1.12pmDID YOU KNOW? It has been 44 years since Everton last won at Turf Moor? Yep, a 2-1 win for the Toffees in March, 1970 was the last time that the Merseysiders have won here. Sure, they've only made the trip a total of five times since that win - drawing three and losing twice - but it's another little omen for the hosts.
1.15pmBurnley won their last Premier League clash with the Toffees here; a 1-0 victory in a game where Wade Elliot netted the winner and Louis Saha missed a penalty. How Dyche would love a repeat of that scoreline. The result is far more important than the performance here today.
1.17pmBENCH WATCH: We can't ignore the quality of Everton's bench compared to Burnley's, however. Such disparity may well be the difference between the sides. With Ross Barkley, Aiden McGeady and Steven Pienaar in the dugout, Everton have plenty of inspiration in reserve if they need it. Do Burnley? Well, not really. Ashley Barnes? Marvin Sordell? Ross Wallace? Hmm. You can make your own minds up.
1.20pmPREDICTION! Right, we're about 10 minutes from kickoff at Turf Moor so let's put our cards on the table. I fancy an Everton win but I reckon it might be a hard-fought three points for them. Burnley, if nothing else, have defended very well at times this season but I think Martinez and co. will nick it in the end. Let's go 2-1 Toffees.
1.23pmGood thing Aiden McGeady is on the bench because it looks like Burnley haven't even given him a seat in the dressing room.
4 minOh dear this is a shocking start for Burnley. All of Dyche's pre-match plans gone to waste so early. It's Samuel Eto'o, who leapt highest to nod a Leighton Baines cross past Tom Heaton via the crossbar. Very, very ominous for Burnley indeed.
5 minGeorge Boyd was perhaps a bit guilty of not tracking Baines down that left-hand side, but it was David Jones who allowed Eto'o to move ahead of him and reach his teammate's delicious cross. Both moments will be disappointing for Dyche.
7 minVery little by way of a response from Burnley so far as Everton continue to boast so much of the possession. They just can't keep it when they do find it.
10 minThis is a fantastic touch from Ings, who plucks Boyd's over-the-top ball out of the sky before squaring to Jutkiewicz, whose cross is over the head of Ings then. Still, promising for the hosts - their only promising attack thus far.
12 minSHOT! Ambitious from Romelu Lukaku, who takes aim from all of 30 yards. It's close, to be fair, but it's just wide.
13 min13 minutes in and I can tell you that Everton boast 72% of the possession thus far. They have the stat that matters, too, courtesy of Samuel Eto'o's bullet header early on.
15 minBurnley prepare to whip in their first corner now after, and it's a fine delivery from David Jones but Eto'o clears for another. This is one isn't too good, however, and goes over everybody.
17 minThis is a good spell from Burnley as they launch a few hopeful balls into the Everton box, which the visitors deal with but there's reasons to be cheerful from a Burnley perspective.
19 minThis is a good move but it's narrowly offside. Ings chests Stephen Ward's clearance into the path of Arfield, who plays a first-time through ball for Jutkiewicz but the striker is narrowly ahead of the last defender. They have to keep this tempo whilst ensuring that they're solid at the back.
21 minAaaand we're back on level terms at Turf Moor folks. But it's a massive error from Lukaku, who needlessly concedes possession to Jutkiewicz deep in Burnley's half. Jutkiewicz then finds Ings, who races through, rounds Tim Howard and slots into an empty net for his first Premier League goal ever. That will do them the world of good.
22 minCLOSE! Burnley tails are well and truly up now as Scott Arfield fires a half-volley just wide of Howard's post! Fair play to them.
24 minBOOKING! Romelu Lukaku goes into Andre Marriner's book for a late challenge on George Boyd to bookend a miserable last few minutes for the Belgian.
26 minEverton haven't really responded to being pegged back and Burnley still marginally the better side.
29 minLukaku atones for his earlier error and restores Everton's lead. It's scrappy, but full marks to Steven Naismith in the build-up to it. Lukaku let the Scot's past run by him before taking aim inside the box; his first shot was well blocked by Shackell, but he fired past Heaton at the second attempt and give his side the lead once more.
30 minBOOKING: George Boyd is booked now after a late challenge on Osman.
31 minDecent enough first half, eh? Burnley, to their credit, look unperturbed to falling behind again and still continue to build promising attacks.
32 minSHOT! Jutkiewicz does really well here, out-muscling Alcaraz near the half-way line before galloping through down the left-wing but, with Ings in the middle, Jutkiewicz elected to shoot and sliced his effort well wide of the mark.
34 minReally good chance for Burnley after Baines concedes a free kick just outside the box for fouling Kightly...
35 minBut it's a poor effort from David Jones, who hits the wall. In his defence, it looked one of them where it might be too close to score.
37 minBurnley need a response. It has gone a little flat since Lukaku's goal, with Jutkiewicz's wild effort the only talking point. They've done well overall, though.
39 minGood work from Michael Kightly, nutmegging Seamus Coleman but his final ball isn't the greatest and Jagielka gets it away. Not many make a mug of Coleman but Kightly did the job there.
40 minOFF THE BAR! Everton might be going into half-time with a two-goal lead if not for the crossbar, as Naismith nods a Leon Osman cross against the woodwork. They might rue that opportunity come 3.30pm.
42 minSamuel Eto'o is receiving treatment following a challenge by Arfield. It's a like a scene from The Grey at the moment, with Burnley fans howling and whistling the veteran striker. Presumably they think he dived but there looked a bit of contact.
44 minCan Burnley get back on level terms again before the break? One minute to find out but Everton should hold out for the interval at least.
45 + 2 minSHOT! Arfield fires a shot narrowly wide, decent effort but it was never on target and that should be that for this half.
2.20pmWhat do we make of that half then? Plenty of positives for Burnley but they still find themselves behind through Romelu Lukaku's goal, after Danny Ings had cancelled out Samuel Eto'o's earlier opener.
2.23pmStill, Sean Dyche won't be too unhappy given what he has seen from this first half, because Burnley have, at times, looked very decent themselves. They looked a bit suspect defensively, however. The second goal looked far too easy, nowhere enough tackles going in to stop Naismith from driving through.
2.26pmIncidentally, you can read a full round-up of the first 45 minutes here!
2.26pmIn the meantime, pop the kettle on and meet me back here in a few minutes for the second 45 minutes. There will be goals. Oh yes.
45 minRESTART! Aaaaaaaaaaaand we're back underway at Turf Moor, my friends. Burnley kick off what ought to be a very decent half of football if the first 45 minutes is anything to go by.
47 minGood start to the half by Burnley, who fizz a couple of dangerous balls into the box early on. Ings and Jutkiewicz aren't the tallest of players, however, so they might want to abandon that tactic.
49 minCHANCE! Antolin Alcaraz is extremely lucky Jutkiewicz isn't all that lethal, because the Everton defender makes a complete hash of a routine clearance, allowing the Burnley man to head the bouncing ball but it's straight at Tim Howard. We did say that Everton conceded more goals through errors than any other side in the league before the game, so that might've been Burnley's chance to follow suit.
50 minSlow start by Everton but they're growing into the clash a little more now. There are definitely goals in this half, ladies and gentlemen.
52 minBOOKING! Scott Arfield becomes the third player to be booked after cynically fouling Leon Osman.
54 minBOOKING: Another booking now as Steven Naismith cynically hauls down Kieran Trippier. Another good decision from Andre Marriner.
56 minStephen Ward has just provided the comedy factor, after the Irishman looked to launch a long throw into the Everton box, but his hands slipped at the last moment and the ball ended up going quite high, but only about a foot away into the feet of Seamus Coleman.
58 minExcellent tackle from Jason Shackell there to cut out Steven Naismith's run after Eto's searching ball found the Scot.
60 minSHOT! Everton register their first meaningful shot of this half as Eto'o, standing on the edge of the area, looks a bend a shot around Heaton but the Burnley keeper collects it comfortably.
62 minJagielka has been excellent this afternoon overall. He puts another fantastic tackle in on Ings, who appealed for foul but nothing was given. He has had to be excellent, too, because Alcaraz has had one or two dodgy moments.
63 minBOOKING: The yellows are piling up now as Stephen Ward goes into the book for a challenge on Seamus Coleman.
63 minCHANCE! And Baines whips in a fantastic cross from the resulting free kick but Tom Heaton did well to push it for a corner, which came to nothing.
64 minSUBSTITUTION: The good news for Burnley? Romelu Lukaku comes off. The bad news? Ross Barkley replaces him. Meanwhile, Ashley Barnes replaces Lukas Jutkiewicz for Burnley.
67 minStill difficult to call this one. Everton obviously have the lead, and might well grab another, but Burnley have looked dangerous at times throughout this half. They need Ashley Barnes to have a better game than Jutkiewicz was having, though.
69 minBurnley fans still in good voice ahead of this last 20 minutes or so but it's not really having too much of an effect as yet. Everton well on top, with Ross Barkley particularly impressive since his introduction.
72 minEverton are winning the passing stakes, anyway; Burnley 279 - 517 Everton. They've also got the only stat that counts, too. Burnley need a moment of magic from somebody but it's anyone's guess who will provide it.
73 minBOOKING: Ashley Barnes now goes into the book for a challenge on Gareth Barry, who looks to be suffering the effects of that tackle. He's receiving medical attention, but he's back on his feet now.
76 minThis one is in danger of petering out unless Burnley find a bit of inspiration from somewhere. They do have a free kick so let's see what this brings...
77 minIt comes to nothing as Jagielka thrashes away. They're being urged on by the supporters here but they look fresh out of ideas so far. 13 minutes remaining.
80 minOh dear. Jason Shackell, a centre-half in his defence, has a chance to swing the ball back into the box following a long throw, but it's a terrible delivery and glides well wide.
82 minSUBSTITUTION: Steven Pienaar is on now after replacing Leon Osman. Burnley surely need to make a change soon. It's all gone a bit stale.
84 minSUBSTITUTION: Just as I type that, Marvin Sordell jogs on to the field to replace to Michael Kightly. He's a striker, and they need a goal so it makes sense. He badly needs to have an impact.
87 minPut a fork in it, folks. Samuel Eto'o bags his second of the game, and it's a stunner. Standing on the edge of the box, Eto'o used Jason Shackell as a shield and bent the ball around him and goalkeeper Tom Heaton. Fantastic goal.
88 minCHANCE! It could, and probably should be 3-2. Ings weaved away from one or two tackles before driving a left-footed shot well over the bar from a really good position. Terrible finish.
90 minCHANCE! Good save from Tom Heaton to deny Steven Naismith, who danced his way into the box and fired near post but Heaton did well to keep it out. Disappointing end to this for Burnley. Really disappointing.
90 + 1 minIncidentally, Scott Arfield has just been announced as man of the match at Turf Moor. Very odd.
90 + 3 minOFF THE POST! This is so, so cruel. Eto'o is so unlucky to be denied his hat-trick, rounding Tom Heaton and tried to squeeze the ball home from the tighest of angle, but it comes off the post and rolls on the goal-line before it is eventually cleared.
3.23pmThat's all she wrote, my friends. Really poor second-half showing from Burnley it has to be said, but credit to Everton nevertheless. Samuel Eto'o grabbed the only goal of the half in fine fashion to put Burnley out of their misery.
3.26pmAnyways, that's all we've got time for, folks. Stay tuned for analsyis and player ratings for this one. If you're not bored of football for one day then join my colleague Jack Prescott for commentary of Manchester United's clash with Chelsea at Old Trafford! You can find that here. Until next time, enjoy your Sunday.