It is the first time in 21 competitive meetings between the two sides that Swansea have come away with the win.
Read how all the action unfolded below:
7.17pmGood evening! 2013 Capital One Cup champions Swansea City host Everton in the third round of this season's tournament. Both sides have named their starting lineups, so let's take a look at the teams...
7.25pmSo Garry Monk makes five changes to the side that were beaten 1-0 by Southampton at the weekend. Wilfried Bony is suspended, so Bafetimbi Gomis starts up front, while Gerhard Tremmel comes in for Lukasz Fabianski between the posts. Richards, Carroll and Montero are also brought into the side, with Rangel, Ki and Routledge all dropping to the bench.
7.28pmThe biggest piece of team news for the visitors is the return for Bryan Oviedo to the starting lineup. The Costa Rica international suffered a double leg break while playing in the FA Cup in January, but is fit enough to make his first start since then today. Roberto Martinez is resting some of his regulars though, with Romelu Lukaku and Kevin Mirallas both dropping to the bench.
7.30pmHere are the visitors arriving at the Liberty Stadium a short time ago:
7.33pmEverton have a good record in their previous meetings with Swansea, with 15 wins and five draws from their previous 20 meetings with the Welsh side. The hosts have had the better start to this season though, as they sit fifth in the Premier League table, while Everton have struggled for form, leaving them 14th.
7.36pmAt least one of these two sides will end up having a better season in this tournament than they did last year. Swansea were knocked out of the League Cup in the third round last year, suffering a surprise 3-1 defeat to Championship side Birmingham City. Everton also failed to make it past this stage last year, having been beaten by Fulham.
7.39pmAs the below tweet from Everton suggests, it's not the nicest weather in Swansea this evening. It's expected that there will be showers throughout the match, getting heavier shortly after kickoff.
Some pretty intense trance music trying to get the crowd pumped ahead of kick-off. Pretty tough on a wet Tuesday night... #SWAvEFC
3 minThere's not been a lot of action in the opening couple of minutes, with Swansea keeping the ball at first, before Distin and Oviedo pass the ball around the back for the visitors. Dyer tried to run past the latter, but the returning defender does enough to prevent Swansea from getting an early sight at the Everton goal.
4 minCHANCE! Gomis and Dyer continue to look for an early chance for the hosts, but they can't find a way through. Sigurdsson wins the ball back on the edge of the box and unleashes a curling shot, but it's wide of the top corner.
6 minEverton find themselves bearing down on the Swansea goal for the first time as Hibbert, who is the team's longest serving player, makes his way forward. After a quick one-two with Atsu, the veteran defender looks to drill the ball beyond Tremmel, but his shot was always rising, and failed to trouble the Swansea goalkeeper.
8 minGomis puts the visitors under pressure again with a quick run into the box, and manages to win the first corner of the night, but Distin is ready to clear as the ball is curled back into the box.
11 minIt's been a pretty open first 10 minutes, with both sides getting plenty of time on the ball, but neither side have been able to test the keepers yet. Samuel Eto'o gets his first real sight of goal, but his first touch is poor and it goes behind for a goal kick.
14 minCHANCE! Atsu and Hibbert link up again down the right flank, with Hibbert looking to cross in for Eto'o. The initial effort is blocked, but the striker picks the ball up 30 yards out before playing a through-ball for Atsu, but the on-loan Chelsea man can't get the better of Tremmel.
16 minCHANCE! Tim Howard is called into action for the first time this evening as Jefferson Montero rounds several Everton players before trying to squeeze the ball in from a tight angle, but the keeper is ready, and comfortable holds onto the shot.
18 minBOOKING! Federico Fernandez picks up the first yellow card of the evening for a wild challenge on Oviedo.
20 minCHANCE! Garbutt curls the resulting free kick in towards Alcaraz, but Tremmel is on hand to deny the defender's header.
22 minCHANCE! Tremmel is starting to see a little more of the ball as Everton have been looking pretty sharp in the last couple of minutes. Besic wins the ball out wide and pulls it back to McGeady, but the midfielder's finish lacks power, giving the keeper a comfortable save.
24 minEverton are definitely looking to more likely of the two sides at the minutes as Besic lines up another shot for the visitors, but he gets his finish all wrong, firing high into the stand behind the goal.
28 minGOAL! Swansea City 1-0 Everton (Nathan Dyer)
28 minAfter a dominant spell from Everton, Swansea finally manage to get the ball back, with Montero cutting inside before crossing for Nathan Dyer, who powers the ball in from just a couple of yards out.
31 minCHANCE! Besic, who has been one of Everton's best players going forward so far, looks for a quick equaliser, creating some space on the edge of the box before trying to pick out the top corner, but it went just wide.
33 minDyer makes a quick run down the right as Swansea look to get back on the attack, but a mis-communication with Sigurdsson sees the ball go out for an Everton throw in.
36 minCHANCE! Everton are immediately involved at the other end as Alcaraz makes a blistering run before pulling it back for Atsu, but Williams deflects his finish behind for a corner. Besic goes for a header from the corner, but the Swansea captain is once again on hand to put it behind, before Tremmel gathers from the third attempt.
39 minMontero battles hard with Hibbert as he tries to get a cross in for the hosts. His first effort is blocked, but he gets another chance. However, he slips while attempting some flashy footwork near the corner flag, putting the ball out of play.
41 minGomis has been kept pretty quiet after a busy start for the striker, and he tries to chase a loose ball in the penalty area, but Howard gets to it first for Everton.
43 minCHANCE! Montero is involved again and Swansea are finishing this half strongly. He curls the ball in from the left, but Hibbert deflects it behind for a corner. Gomis tries to get a header away as the ball is curled back in, but he doesn't rise high enough to meet the cross.
45+1 minGomis runs at the Everton back four, but Oviedo does enough to block the forward's progress. Shelvey wins it back in midfield, but can't do anything with it. Meanwhile, there's just a minutes of stoppage time to be played.
8.33pmSo Nathan Dyer's 28th-minute strike has given Swansea a 1-0 lead over Everton at half time in the third round of the Capital One Cup, leaving the South Wales side in line for their first ever win over the Toffees.
8.37pmLet's take a look at some of the stats from the opening 45 minutes. Everton just edged the possession with 51% and have had eight shots so far, three more than their opponents, although both have had two on target. Swansea have committed five fouls compared to Everton's three, and saw the only booking of the first half.
8.41pmRoberto Martinez won't be panicking too much after his side fell behind. The visitors have plenty of fire power on the bench, with the likes of Romelu Lukaku, Kevin Mirallas and Arouna Kone available to come on. All of the Everton substitutes are out warming up while the rest of the team are in the dressing room.
8.45pmThe players are just starting to make their way back out for the second half.
8.46pmSUB! Everton are making a change before the action gets back underway, with Lukaku replacing Eto'o.
46 minKICKOFF! Everton get the second half underway.
46 minCHANCE! The substitute striker almost has an immediate impact as he skips around Fernandez before launching a powerful shot, but it finishes just high of the crossbar.
49 minSwansea win the ball back and try to put the visitors under pressure again. Initially Montero looks to deliver the ball, but his path is blocked, before a weak effort from Shelvey is easily claimed by Howard.
51 minCHANCE! Swansea still have the ball and Montero is involved again, but his cross is blocked by Alcaraz. The winger quickly wins it back and plays it in for Sigurdsson, but his header is punched clear by Howard.
54 minSince Lukaku unleashed a strike in the first minute of the second half is really has been all Swansea. Gomis and Richards link up down the right before Dyer tries a cross, but it's deflected behind for a corner.
56 minSigurdsson curls the corner in, but Lukaku is back to help out the defenders and heads it clear. The visitors try to go on the counter-attack, but Tremmel comes out of his area to boot it away for Swansea.
58 minSUB! Everton make their second change of the night, and it's Oviedo going off, having played almost an hour in his first match in eight months. James McCarthy takes his place.
60 minBOOKING! The second booking at the night goes to Jonjo Shelvey as he clashed with McCarthy in midfield, pulling the midfielder to the ground. McCarthy appears to have given Everton a bit more pace since replacing Oviedo a couple of minutes ago.
62 minAtsu and Besic try to find a way through for the visitors, who really haven't created much in this second half, but the ball is cleared before it gets to Lukaku.
64 minMontero is on the ball again for Swansea, and wins a throw near the corner flag after battling with Hibbert. Gomis chases as the ball is delivered into the box, but the striker can't get a shot away.
65 minGOAL! Swansea City 2-0 Everton (Gylfi Sigurdsson)
65 minIt's been coming for the majority of the second half, and the home side have now doubled their lead. Jonjo Shelvey curled a well-placed cross into the box, which was initially headed into the crossbar by Distin as the Everton defender tried to clear the danger, but it fell to Sigurdsson, who was left with a simple finish into the bottom right corner. It feels like it would take a big effort for Everton to get themselves back into this tie now.
68 minSUB! It's a first change for the hosts, and it's the scorer of the opening goal who makes way, with Wayne Routledge replacing Nathan Dyer.
70 minEverton have responded well since conceding the second, with Gibson and Lukaku both working hard to try and set up a chance for Everton, but Swansea seem to have switched to a more defensive set-up, and it stops the visitors from testing Tremmel.
71 minSUB! The hosts make a second change, taking off Tom Carroll for Ki Sung-Yeung.
74 minAfter a bright spell from Everton, Swansea are back in control, but their unable to add a third as an impatient Shelvey launches a 30-yard strike well wide of the Everton goal.
77 minEverton looks as though they have run out of ideas as Swansea effortlessly keep the ball at the back before Neil Taylor brings it out before looking for Montero. Distin eventually gets it back for Everton, but the hosts keep them pinned back inside their own half.
79 minMcCarthy picks out Lukaku with a brilliant through-ball, but Tremmel rushes off his line to strip the ball away from the Belgian international.
80 minEverton are starting to see a bit more of the ball again as McGeady makes a run down the left flank before beating Richards to get a cross in, but Williams is on hand to boot it clear.
81 minSUB! The visitors make their final change, with Leon Osman coming on for the final 10 minutes in place of McGeady.
83 minLuke Garbutt floats a dangerous-looking cross into the box for Lukaku which leaves Tremmel scrambling, but the German keeper manages to beat the striker to it, punching well clear.
84 minCHANCE! Shelvey tries another chance from range, but this time it takes Howard to keep the powerful 30 yard drive out.
87 minGOAL! Swansea City 3-0 Everton (Marvin Emnes)
87 minSubstitute Marvin Emnes takes less than two minutes to make an impact on the match as he weaves his way through the Everton defender before blasting a low effort past Howard, surely ending any hopes of a fightback from the visitors.
89 minEverton win a free kick in a dangerous position, which Garbutt curls in towards Osman, but it's put behind for a corner by Williams, before the resulting corner is swiftly dealt with by the hosts.
90 minCHANCE! Well Everton certainly haven't given up hope of a late goal as Gibson has what is probably the Toffees' best chance of the night, but his low effort is smashed just wide of the left upright.
90+3 minAs the final of three minutes of stoppage time begins, Lukaku tries to set up Atsu for a consolation goal, but his through-ball has too much pace and goes straight through to Tremmel.
90+3 minBOOKING! Luka Garbutt picks up a late booking for a clash with Emnes on the right sideline.
9.37pmSo the 2013 League Cup winners progress to the fourth round with what turned out to be a comfortable win over Everton. It's there first ever competitive victory over the Toffees.
9.42pmSwansea will have to wait until tomorrow to find out who they will play next, as that's when the fourth-round draw takes place. So that's all from the Liberty Stadium this evening, but stick around to read about all of tonight's results in the competition, and join us again tomorrow for the remaining third-round ties and to find out the results of the draw. Until then, thanks for joining me this evening, goodbye.