The Swans booked their place in the Capital One Cup fourth after a convincing 3-0 win over Everton at the Liberty.
Goals from Nathan Dyer, Gylfi Sigurdsson and substitute Marvin Emnes sealed a first ever victory over Everton as the Swans ended a run of two consecutive defeats.
Dyer scored his fourth goal in seven games to put the Swans ahead, guiding home a cross from the excellent Jefferson Montero after 28 minutes.
Sigurdsson then doubled the lead - and his season's tally - in the 64th minute after he bundled home after Sylvain Distin's header deflected over keeper Tim Howard and off the crossbar before dropping into the patch of the midfielder.
And sub Emnes wrapped things up late on as he fired low into the corner just two minutes after having being introduced.
The Swans made five changes to the starting XI.
Gerhard Tremmel, who kept a clean sheet in the previous round against Rotherham, god the nod between the sticks while Jazz Richards was the one change in defence.
Loanee Tom Carroll was brought into midfield while Jefferson Montero started on the wing in place of Wayne Routledge, who moved to the bench.
The final change saw Bafe Gomis, who scored the winner against the Millers in round two, lead the line with Wilfried Bony serving his one-match suspension.
Everton boss Roberto Martinez made seven changes to his line-up as Bryan Oviedo made his first start since breaking his leg in an FA Cup tie against Stevenage last January.
Tony Hibbert and Antolin Alcaraz also went into the Toffees defence while Darron Gibson, Aiden McGeady, Muhamed Besic and Luke Garbutt made up the other changes.
SWANS: Gerhard Tremmel, Jazz Richards, Federico Fernandez, Ashley Williams (capt), Neil Taylor, Tom Carroll (Ki Sung-Yueng, 70), Jonjo Shelvey, Gylfi Sigurdsson (Marvin Emnes, 85), Nathan Dyer (Wayne Routledge, 68), Jefferson Montero, Bafe Gomis.
SUBS NOT USED: Lukasz Fabianski, Angel Rangel, Kyle Bartley, Jay Fulton.
EVERTON: Tim Howard, Tony Hibbert, Darron Gibson, Samuel Eto'o (Romelu Lukaku, ht), Aiden McGeady (Leon Osman, 81), Bryan Oviedo (James McCarthy, 58), Sylvain Distin, Muhamed Besic, Christian Atsu, Luke Garbutt, Antolin Alcaraz.
SUBS NOT USED: Joel Robles, Arouna Kone, Kevin Mirallas, Tyias Browning.
Both teams created chances within the opening ten minutes as Gylfi Sigurdsson saw two efforts drift wide of goal while Tony Hibbert and Samuel Eto'o were off target for the visitors.
Neat footwork from Eto'o almost sent the lively Christian Atsu in on goal, but the onrushing Gerhard Tremmel beat him to the ball as he gathered low down.
And Everton went closer again as an inswinging free-kick fell to Antolin Alcaraz in the box, and the centre-back was denied at point-blank range from Tremmel as the German keeper deflected it out for a corner.
On this damp evening, the sides cancelled each other out until the 28th minute as Jefferson Montero's pace caused havoc down the left and when his low cross fizzed along goal to Nathan Dyer, the winger's low shot was parried into the bottom corner by Tom Howard.
Everton went in search of an equaliser, and it took a brave challenge from skipper Ashley Williams to deny Alcaraz as he beat the Toffees centre-back to a cross.
Then, up the other end, Montero again worked his magic down the left but his cross just evaded the head of Bafe Gomis, who was under pressure from Sylvain Distin.
And Montero again probed, this time his shot looping over Howard and out for a corner following a deflection off Hibbert.
The Swans went close from the resulting corner, Williams' header from Sigurdsson's delivery flying across the face of goal.
That was it for the first half, with the Swans deservedly taking a 1-0 advantage into the break.
HALF-TIME: Swans 1 Everton 0
Roberto Martinez made a change at the break with Romelu Lukaku replacing Eto'o in a straight swap up front.
And the Belgian settled straight into the action as he latched onto Bryan Oviedo's pass before firing over Tremmel's crossbar.
Aiden McGeady then tried his luck from 20 yards, but, like Lukaka, his effort failed to trouble the Swans keeper.
The Swans, though, hit back via Sigurdsson. After Montero latched onto Jonjo Shelvey's threaded pass, the Ecuadorian's cross found Sigurdsson, but his header was swatted off the line by Howard.
Martinez made another change, this time James McCarthy replacing Oviedo.
But it was the Swans who made the next big breakthrough, albeit it in rather bizarre circumstances after 64 minutes.
When Shelvey's cross bounced in the area, Distain's attempted diving header clearance looped over Howard and rebounded back off the crossbar. Sigurdsson was quick to react, and the Icelandic international bundled home from close range.
Garry Monk then made two changes in quick succession as Wayne Routledge and Ki Sung-Yueng replaced goalscorer Dyer and Tom Carroll.
Tremmel then had to be alert as he raced off his line to deny Lukaku, who was in on goal after Shelvey's pass deflected off Besic and into the striker's path.
Martinez made his final change inside the final ten minutes as Leon Osman replaced McGeady with the visitors looking to claw themselves back into the game.
But the Toffees had to survive a penalty appeal as Routledge's cross struck the arm of Luke Garbutt, but referee Roger East waved away the home claims.
The home side continued to look to add to their lead, and Shelvey went close to doing that but his thunderous 20-yard left-footed strike was padded wide for a corner by Howard.
Marvin Emnes was then introduced in place of scorer Sigurdsson for the remaining five minutes, as Monk made his final change.
And Emnes made an immediate impact, collecting Shelvey's pass before racing on and firing low past Howard and into the bottom corner just two minutes after having come on.
That was the end of the action, with the Swans winning in dominant fashion.