On-loan striker Lacina Traore made the perfect start to his Everton debut but it was substitute Steven Naismith who played the pivotal role in their FA Cup victory over an under-strength Swansea.
Traore, who joined in January from Monaco, scored after just four minutes but it was his second-half replacement who secured the 3-1 win and safe passage into the quarter-finals.
The Scotland international scored within four minutes of coming on to restore Everton's lead following Jonathan de Guzman's equaliser and then won the penalty which Leighton Baines converted.
Toffees manager Roberto 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 - apart from possibly the added length of his legs - to do with the goal.
Wayne Routledge's foul on Ross Barkley gave Kevin Mirallas the chance to swing in a free-kick from the left and, although Sylvain Distin fluffed his header, he was given a second chance by Kyle Bartley and calmly picked out Traore.
He swung one of those long legs at the ball and generated plenty of purchase to send his backheeled 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.
Whether the unusual sight of the sun glaring down at a low angle over the Park End impacted on Everton's defending was open to question, but there is no doubt they did not spot De Guzman's run for the equaliser, having previously allowed Alvaro Vazquez to race through and round goalkeeper Joel Robles but fail to get off a shot into the open goal after two minutes.
Left-back Neil Taylor's cross picked out the run of the Dutchman, who charged from the edge of the penalty area between Phil Jagielka and Baines to powerfully head beyond Joel Robles.
Such was the sparseness of the following support - a couple of hundred at best - the goal barely registered, but it was De Guzman's first for four months and Swansea's first at Goodison Park since 1983.
Slack defending 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 Gerhard 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, Vazquez lobbed a ball over Robles and wide of the far post after Distin - possibly still troubled by that low sun - misjudged a ball over his head.
Barkley, who is still not 100 per cent recovered from a broken toe, 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 backpass exposed Tremmel.
Naismith was then bought down by Ashley 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.