The Spaniard is bidding to become the first manager to win the competition in consecutive years with two different clubs after his success with Wigan Athletic last term.
And he took another step closer to that feat with progression to this year's quarter-finals at Goodison Park on Sunday.
Monaco loanee Lacina Traore marked his Everton bow by opening the scoring in the fourth minute, before Jonathan de Guzman levelled matters 11 minutes later.
Substitute Steven Naismith restored the hosts' lead following a defensive lapse from Neil Tayor in the 65th minute, and Leighton Baines made sure of the win from the penalty spot shortly after.
Swansea came into the match knowing that, to reach the last eight for the first time since 1964, they would need to beat Everton for the first time in their history and score a first goal at Goodison since 1983.
However, head coach Garry Monk's team selection suggested he had one eye on Thursday’s UEFA 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 throughball 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 debutant Traore - a towering presence in the Swansea penalty area - cleverly flicked home from close range after some good work by Sylvain 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 Gerhard 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.
But as the half progressed, it began to look increasingly likely that the sides would need to meet again at the Liberty Stadium to determine a winner.
However, after Pienaar had sent a powerful header wide of the left-hand upright, Naismith capitalised on Taylor's misplaced back pass to slot the ball underneath Tremmel and into the back of the net.
And Everton could feel sure of their place in the quarter-finals seven minutes later as Baines converted a penalty after Naismith had been felled by Jazz Richards.
