The Welsh club are winless in eight games and sit four points above the drop zone following a 3-2 defeat to Martinez's Everton at Goodison Park on Saturday.
Swansea head coach Garry Monk enjoyed a 3-0 success over rivals Cardiff City in his first game after succeeding Michael Laudrup in February, and also oversaw two encouraging performances in a UEFA Europa League last-16 tie that went in Napoli's favour.
However, Swansea have not tasted victory since their derby triumph and face a battle to retain their top-flight status.
Martinez, who played for Swansea between 2003 and 2006 before returning to manage the club within 12 months, has backed his old side to stay clear of the bottom three.
Reflecting on the challenge the Liberty Stadium outfit have faced due to European commitments, the Spaniard said: "They have had a really different season - the experience in the UEFA Europa League has cost them a few points but they will take that. They are a better side after being involved (in the competition).
"I am certain they will get enough points to achieve their aim this season and their football principles are so strong that they will be able to get wins in the next few games."
Swansea face a tough away game on Tuesday against an Arsenal side looking to bounce back from a humiliating 6-0 defeat to title rivals Chelsea.
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