The hosts raced into an early lead, as Sylvain Distin kept a set piece alive and squared for the towering Monaco loanee, who coolly backheeled the ball into the back of the net from close range.
Swansea hit back almost immediately, with Neil Taylor whipping in a pinpoint cross that was powered beyond Joel Robles by Jonathan de Guzman.
In an error-strewn match it was perhaps inevitable that a mistake produced the next goal, as Taylor's back pass to Gerhard Tremmel was intercepted by Steven Naismith, who calmly slid the ball beneath the stranded German keeper.
Naismith then had a hand in the Toffees' third goal, tripped by Ashley Richards inside the box, and Leighton Baines duly stepped up to bury the penalty and book Everton's passage to the sixth round.
Swansea came into the match knowing that, to reach the last eight for the first time since 1964, it would need to beat Everton for the first time in its history and score a first goal at Goodison since 1983.
However, coach Garry Monk’s team selection suggested he had one eye on Thursday’s Europa League clash with Napoli, while opposite number Martinez named a near full-strength side, handing a debut to Traore.
Despite the weakened starting XI, Swansea went close to breaking the deadlock early on as Pablo Hernandez played a wonderful through ball into the path of Alvaro Vazquez, but the striker failed to make the most of the opportunity.
And the visitors were punished moments later, when Traore — a towering presence in the Swansea penalty area — cleverly flicked home from close range after some good work by Distin.
But Swansea pulled level in the 15th minute, as De Guzman stormed into the box with a perfectly timed run to power home a header from Taylor's cross.
Steven Pienaar then hit the crossbar as Everton looked to regain the lead, before Tremmel saved well when Traore got his head to the rebound. Kevin Mirallas almost curled home a 20-yard free kick five minutes from the interval, but the German goalkeeper was again on hand to punch clear with a diving save.
Everton started the second half brightly, and Ross Barkley pulled the trigger from the edge of the penalty area with an effort that stung the palms of Tremmel.
Then, after Pienaar had sent a powerful header wide of the left-hand upright, Naismith capitalized on Taylor's misplaced backpass to slot the ball underneath Tremmel and into the back of the net.
And Everton could feel sure of its place in the quarterfinals seven minutes later as Baines converted a penalty after Naismith had been felled by Richards.
The result keeps Martinez's hopes of retaining the FA Cup alive as he bids to become the first manager to win the competition in consecutive years with two different clubs after emerging victorious with Wigan last season.