The two clubs first played each other in September 1930 but Swansea had failed to win in 20 attempts ahead of this tussle.
Yet goals from Nathan Dyer, Gylfi Sigurdsson and Marvin Emnes comfortably saw them win through to the fourth round at the Liberty Stadium on Tuesday night.
Dyer opened the scoring with his fourth goal of the season after 28 minutes, Sigurdsson added another midway through the second half and substitute Emnes drove home the third in the closing moments.
Everton had made seven changes to the side beaten by Crystal Palace in the Barclays Premier League on Sunday, and Swansea five to the team which lost to Southampton a day earlier, but there was no shortage of quality on the pitch and plenty of strength on both benches.
Bryan Oviedo made a welcome return for Everton after suffering a double leg break in an FA Cup tie at Stevenage in January, but it was Swansea winger Jefferson Montero who caught the eye with a fleet-footed display down the left flank.
Samuel Eto’o dragged horribly wide from a clear opportunity and, though both sides were comfortable in possession, chances were scarce in an opening period which Swansea slightly shaded.
Everton were showing signs of growing into the game when Luke Garbutt’s deep free-kick caught Swansea sleeping and Antolin Alcaraz stabbed the ball towards goal, with only Gerhard Tremmel’s legs preventing the visitors from taking the lead.
It proved a precious save as Montero surged between Hibbert and Muhamed Besic in the 28th minute and crossed into the danger area.
Bafetimbi Gomis failed to make contact but the ball travelled through to Dyer at the far post to beat Tim Howard with a crisp sidefooted finish, his fourth goal of the season.
Everton rallied for Sylvain Distin to send a header straight at Tremmel but Ashley Williams also provided a corner threat with his effort flashing across Howard’s goal.
The Toffees sent on Romelu Lukaku for Eto’o at half-time and the big Belgian striker almost made an instant impact by outmuscling Federico Fernandez to hit a rising shot over the crossbar.
Aiden McGeady also fired over but Swansea almost doubled their lead after 53 minutes with Montero again the provider following a defence-splitting pass from Jonjo Shelvey.
The Ecuadorian picked out Sigurdsson with a delicious cross and Howard had to save acrobatically to stop the header from beating him.
But Howard had no chance after 64 minutes when Distin headed against his own crossbar in trying to clear and Sigurdsson reacted quickest to stab home the loose ball from close range.
Howard denied Shelvey but Emnes fired home 74 seconds after arriving as an 85th-minute substitute as Swansea, winners of this competition two seasons ago, eased through.
Victory was particularly sweet for rookie Swansea boss Garry Monk who was up against his great friend and former manager, Roberto Martinez, in the Everton dug-out.