The hosts then dominated the second period at Goodison Park with substitute Naismith scoring in the 65th minute, soon after replacing Traore.
The Scotland international then won a penalty seven minutes later when he was brought down by Ashley Richards and it was converted by Leighton Baines, the England defender's 15th successful spot-kick for Everton in 16 attempts.
Everton will travel to Arsenal in the last eight after Arsene Wenger's team beat Liverpool 2-1.
"We're very satisfied because that's all you're looking for - a win to get through to the next round," former Swansea boss Martinez told the club's official website (www.evertonfc.com).
"I thought we were a little bit anxious (in the first half). I think we were feeling the role of playing at home and the expectation to get through to the next round and we were a little bit too cagey with our possession.
"We allowed Swansea to get really well organised and cause problems on the counter," added Martinez who was in charge of Wigan Athletic when they won the competition last season.
Naismith, who has scored four goals in his last five matches, was forced to leave the pitch late on after receiving a blow to the head.
"Steven's okay. We're going to check him in the next 24 hours. He had a bit of delayed concussion," said Martinez.
"When he had the knock he was okay, then he became a bit dizzy and he couldn't remember much of what was happening. At that point we couldn't take any risks."
(Reporting by Michael Hann, editing by Josh Reich)
