"It was a difficult game and you could tell what was at stake for both teams," Martinez told BBC Sport.
"From our point of view, to keep a clean sheet in those circumstances was very pleasing and it was also important to take one of our chances too. It makes for a very satisfying performance full of character and personality.
"We came into this game on the back of a stylish win at home to Arsenal, but in this league you have to adapt quickly to playing different teams and doing what you have to do."
Everton host Crystal Palace in the Premier League on Wednesday.