The Toffees will be away to Arsenal in the sixth round on the weekend of 8/9 March.
Lacina Traore got his Everton career off to the perfect start, netting a debut goal after just four minutes when he adjusted his feet to back-heel the hosts into the lead from close-range after Sylvain Distin found him inside the area.
Roberto Martinez's side were in front for just over 10 minutes as Jonathan de Guzman equalised with a well-timed header after ghosting into the area to nod Neil Taylor's cross past Joel Robles.
Second-half substitute Steven Naismith restored Everton's lead when he raced onto Taylor's short back-pass before slotting past the onrushing Tremmel.
Naismith was then involved in the third as he was hacked down in the area by Jazz Richards and Leighton Baines emphatically dispatched the resulting spot-kick to seal their progress.
Martinez, facing the club over whom he had such an influence between 2007 and 2009, remains on course to complete his quest of becoming the first manager to retain the cup with different clubs, having lifted it with Wigan last May.
Much has been made of Ivory Coast international Traore's 6ft 8in frame, but being the tallest outfield player in the Premier League had little to do with the goal.
Kevin Mirallas swung in a free-kick from the left and, although Distin fluffed his header, he was given a second chance by Kyle Bartley and calmly picked out Traore, who generated plenty of purchase to send his back-heeled effort flying into the net.
With Swansea's weakened team - showing eight changes from the midweek draw with Stoke - that should have been the signal for Everton to go for the jugular, but it seemed to a man they were all a little off the pace.
Mirallas cut in off the right flank to set up a perfect shooting opportunity, but his final touch allowed the ball to run too far ahead of him and Swansea cleared.
Everton did not spot de Guzman's run for the equaliser, having previously allowed Roland Lamah to race through and round goalkeeper Joel Robles but fail to get off a shot into the open goal after two minutes.
Taylor's cross picked out the run of the Dutchman, who charged from the edge of the penalty area to powerfully head beyond Robles for his first in four months and Swansea's first at Goodison Park since 1983.
Open defence was the trait of the first half, but Traore could not take advantage when Steven Pienaar's deflection from Baines' cross rebounded to him and his weak header was saved by Tremmel.
The loss of Bartley to a hamstring injury on the half-hour saw the reassuring presence of Ashley Williams restored to the back four, but that still did not prevent Mirallas whipping in a free-kick for Tremmel to save and Barkley drilling a low shot just wide.
In between that, Alvaro Vazquez lobbed a ball over Robles and wide of the far post after Distin - troubled by low sun - misjudged a ball over his head.
Barkley, who is still not 100 per cent recovered from a broken toe last month, had another shot saved in the second half before he and Traore were replaced by Leon Osman and Naismith.
Martinez's instincts proved correct as, moments after Pienaar planted a simple header wide, Naismith scored his fourth in five games after Taylor's back-pass exposed Tremmel.
Naismith was then bought down by Richards, allowing Baines to extend his impeccable record from the penalty spot in the 72nd minute.
Everton finished the match with 10 men as, having used all three substitutes, Naismith went off after trying but failing to carry on after being caught in the head by Jordi Amat, but the hosts were able to comfortably cope with their reduced number.