Keane headed home on the half-hour mark for his first goal in a Toffees shirt, before Gueye produced a neat finish after running onto Wayne Rooney's perfectly-weighted pass.
Hajduk had opportunities of their own after the break but after Jordan Pickford kept the Croatian outfit at bay, Everton were able to celebrate keeping another clean sheet ahead of next week's second leg.
Read below to see how the action unfolded at Goodison Park.
7.09pmDue to their activity in the transfer market, Everton have barely been off the backpages all summer but it is now time for the Toffees to do their talking on the pitch. Breaking into the top six of the Premier Leaguu remains their aim for this campaign, but Ronald Koeman and Everton's supporters are desperate for the club to enjoy a run in Europe.
7.12pmTheir participation between now and Christmas all depends on how they fare in two matches against Hajduk Split. On paper, Everton are justifiably the favourites but they would be naive to dismiss their opponents, who have already come through two testing ties with Levski Sofia and sit second in the Croatian top flight after five matches. There were tougher draws that Everton could have been given, but not many, and the difficultly is cranked up a notch with needing to travel to Croatia for the second leg.
7.13pmBefore we continue with our buildup to this match, let's bring you some team news. How does Koeman set up with a trip to Manchester City coming up on Monday night?
7.23pmKoeman has opted to make three alterations to the team which defeated Stoke City at the weekend, with Cuco Martina, Kevin Mirallas and Ademola Lookman all coming into the starting lineup. It also sees Everton revert to a 4-2-3-1 formation after previously using a 3-4-1-2 tactic. Wayne Rooney leads the line with Sandro Ramirez missing through injury. New signing Gylfi Sigurdsson must wait to make his debut after joining from Swansea City.
7.32pmHajduk are without both former Leicester City forward Marko Futacs and former Barcelona player Borja Lopez, as well as Ante Erceg, but boss Joan Carrillo is still able to name a strong team for their trip to Goodison Park. Franck Ohandza is given the nod in attack after starting the season with two goals from five games, while youngster Nikola Vlasic has netted the same from midfield.
7.38pmThis is where we would usually talk about the previous meetings between the two sides, but it comes as little surprise that these clubs have never met in European competition.
7.41pmOn paper, Everton should progress over the two legs but it is worth noting that both Southampton and West Ham United have slipped up at this stage of the competition in recent years. They came unstuck against teams from Denmark and Romania, and Everton's tie is arguably tougher than what their Premier League rivals had in front of them.
7.44pmWhile Everton possess players with superior ability, they will not be on level terms with the sharpness of their opponents. The English outfit have played three competitive fixtures, but Hajduk Split have nine meaningful games in their legs, with five matches coming in the league and four in Europe. It is something which Koeman will be mindful of.
7.48pmSigurdsson may not feature in Everton's squad for tonight's game, but the Icelandic playmaker will be unveiled to the crowd before kickoff. No-one will criticise Koeman for purchasing Sigurdsson because his record speaks for itself, but you have to wonder where he fits in this team and what it means for his new teammates. Klaassen or Rooney will likely expect to play in the number-10 role, but Koeman has not spent £45m on a player to sit on the bench.
7.51pmAs it stands, there will be only one Premier League side who will be competing in the Europa League. That is Arsenal, but there could still be potentially three clubs competing on a Thursday night. While Everton will be doing their utmost to qualify for the group stages, local rivals Liverpool are doing all they can to stay out of them as they attempt to qualify for the Champions League. Manchester United would have been in this competition had they not emerged victorious in the final in May.
7.54pmPREDICTION! Hajduk have kept clean sheets in three of their last five games and we are expecting them to prove to be a tough nut to crack tonight. Everton have won each of their three games by a 1-0 scoreline since returning to competitive action and we don't expect them to score many this evening. We are going to sit on the fence and predict a 1-1 draw.
8.02pmIt is time for the two teams to make their way out at Goodison Park. It's fair to say there are more supporters inside the ground than there were for the previous round when work was being carried out in one of the stands. Kickoff is coming right up.
3 minIt has been a strong start from both sides. Everton had the early possession but Hajduk have wasted no time in pressurising their hosts into mistakes. They certainly haven't come to sit back.
5 minThis is an astonishing decision. Rooney breaks clear of Nizic down the left flank and the defender pulls him down with the ball long gone. Unbelievably, the referee decides against handing him a yellow card.
6 minCHANCE! This is only a half-chance at best, but Rooney heads a Mirallas corner wide of the post from the penalty spot. The returning hero has began very well.
7 minCHANCE! Another half chance for Everton as the ball falls invitingly to Klaassen who is standing just inside the penalty area. However, he lashes his volley high and wide when goalkeeper Stipica should be forced into some work.
10 minEverton are setting a really high tempo, something which has been missing from their early games this season. They are yet to register a shot on target but it is only a matter of time.
11 minCHANCE! Everton switch off and it allows Hajduk to mount an attack down the left. The ball eventually falls to Vlasic, who drags a low effort wide of the near post. He should be doing better from there.
14 minFair play to Hajduk - they have come right back into this game. Everton are trying to put them under pressure, but they are being told in no uncertain terms that they dare not take the Croatian outfit lightly. It's far more open than I expected.
18 minGood response from Everton, who decide that possession is the order of the day to discourage Hajduk from coming forward. Let's lots of passes going from flank to flank, with Martina and Baines heavily involved in the Hajduk half.
19 minBOOKING! Schneiderlin deservedly goes into the book for a studs-up challenge. He is a bit fortunate not to receive a red.
23 minThat shot on target hasn't come as easily as I predicted. Everton continue to pass the ball about neatly but aside from a couple of low crosses into the area, they are finding it difficult to create a significant opening.
27 minThere's not much else to report I'm afraid. It's all huff and puff from Everton and you can see why Koeman is desperate to bring in a like-for-like replacement for Lukaku. Baines has just managed a miscued volley from 30 yards out but the ball went over the crossbar.
30 minEverton take the lead! It's a brilliant goal too as Baines is released into the penalty area, helped by Klaassen, and after the left-back floated the ball towards the back post, Keane was there to head into the net from close range for his first goal in a Toffees shirt.
35 minHajduk are on the attack but it has all kicked off in the visiting end. Lots of Hajduk supporters are throwing objects and kicking advertising boards. I also saw an Everton ballboy having to be pulled away from the situation. There are going to ramifications for this. The players have been ordered to go and wait by the dugouts.
37 minThe players are moving back into position. Hopefully that is the end of that.
40 minRooney has the ball in the net but he made his run a touch too early. Good decision from the assistant. There are still bits of trouble in the Hajduk end. A fine is coming their way.
43 minBOOKING! There's no difference between this foul and the one on Rooney during the early stages of the game, but on this occasion, the yellow card comes out for Memolla, who pulls down Lookman.
45 minEverton double their advantage and it is another wonderful goal. Klaassen dispossesses a defender before teeing up Rooney. The forward can't get a shot away but plays in Gueye, who advances into the penalty area before flicking the ball into the bottom corner of the net.
45 minThere will be five minutes of added-on time.
45+4 minSHOTS! Hajduk have started to get some shots on targets, but efforts from Vlasic and Barry are struck straight at Pickford.
45+5 minCHANCE! It should be three. Rooney is played in on goal but his effort is too predictable and Stipica guesses correctly to go down to his left and make the save.
8.57pmThe half-time whistle goes at Goodison Park and Everton deservedly lead thanks to goals from Keane and Gueye. Rooney should have made it three just before the referee blew his whistle for the break, but it is so far so good for the Toffees.
9.10pmKoeman has decided to make a change, with Tom Davies replacing Schneiderlin.
48 minLots of possession for Everton at the start of this second half, but Hajduk have still had one speculative effort on goal.
49 minCHANCE! Another chance for Rooney, who heads wide from around 10 yards out after meeting a Martina cross. The forward should at least be hitting the target.
52 minThis is really clever from Lookman, who buys a free kick out of the defender to relieve some pressure on his side. Hajduk are having the better of these early stages.
56 minMore defending for Everton to do after Baines conceded a free kick out on the left but Martina heads the ball clear. The Croatian side are certainly not out of this. Everton need a clean sheet or a third.
60 minFor someone of his ability and quality, you would think that Rooney would be wise to getting caught offside but he has gone too early on several occasions since the restart. There's no need for it really when he is unlikely to have the pace to run through on goal.
61 minCHANCE! How hasn't the referee given a foul here? He waves play on after Gueye is brought down and seconds later, Ohandza has the the opportunity to find the bottom corner from inside the penalty area but Pickford makes the save down to his left.
62 minSUBSTITUTION! A second change for Everton as Koeman replaces Klaassen with Dominic Calvert-Lewin. Rooney had dropped back into midfield.
66 minSUBSTITUTION! It is time for Hajduk to make two changes, with Ante Erceg and Said Ahmed Said being introduced.
68 minEverton need to offer a bit more going forward and Davies is trying to get into the penalty area more often than Schneiderlin, who he replaced. He wins a corner, but Williams sends a weak header wide of the post.
72 minCHANCE! Hajduk continue to be the better of the two sides during the second half and they come close to halving the deficit. Barry has been excellent in their midfield abut on this occasion, he fires the ball just wide from 20 yards.
76 minSUBSTITUTION! A final switch for Everton sees Muhamed Besic replace Mirallas.
80 minJust the 10 minutes to go at Goodison Park. This is a massive 10 minutes in regards to the tie. Everton will take this but if they concede, it is very much game on in Croatia.
82 minCHANCE! Everton are very much on the back foot. A mistake from Rooney leads to an opportunity for Erceg but Pickford makes a one-handed save at his near post.
83 minSAVE! What are Everton playing at? They are retreating minute by minute and Memolla has all the time in the world to shoot on goal from 25 yards, with Pickford being required to pull off another one-handed save.
85 minSUBSTITUTION! A final change for Hajduk sees Toma Basic come on for Gentsoglou.
88 minBetter from Everton, but they twice fail to make the most of crossing positions through Lookman and Baines. Their end product has been poor during this half.
90 minThere will be just two minutes of added-on time.
90+1 minBOOKING! Besic takes one for the team and commits a cynical foul on the halfway line.
10.04pmThe full-time whistle goes at Goodison Park and Everton have held on to register an excellent win over Hajduk Split. They relied on Pickford to keep their opponents at bay at times, but they were good value for their victory. However, with a hostile atmosphere expected next Thursday, this tie is far from over.
10.05pmThat will be all from Sports Mole for this match. Thank you for joining us this evening and we hope to see you again soon. Goodbye for now.