Sebastian Larsson had the hosts on course for back-to-back top-flight wins with a trademark 67th-minute free-kick.
But Leighton Baines finished emphatically from the penalty spot nine minutes later after Connor Wickham brought down Seamus Coleman to earn a share of the spoils.
"I am disappointed to have dropped two points but pleased with the way we reacted once we went 1-0 down," said Martinez, whose team are now unbeaten over six matches in all competitions.
"We played a well-organised Sunderland side and that first goal was going to be decisive. Once they get it, I was extremely pleased with the reaction we had.
"The character we showed, the understanding, the know-how was great. We didn't panic and we ended up having a full-back in the box earning the penalty."
Martinez observed that those fighting qualities made up for a lack of cutting edge as Everton emerged from the first half with nothing to show for some controlled spells of possession.
"I thought we started really well," he said. "We had a real intent and we controlled the game with ease for long spells.
"But we didn't use that control to create clear-cut opportunities. Sometimes the final ball wasn't there, or it was the wrong run, and it didn't allow us to score the first goal that I felt we needed."
Scans allayed fears that Everton midfielder Gareth Barry had suffered a broken leg from an early challenge by Jordi Gomez - a player Martinez felt was fortunate to remain on the field after going to ground easily under a challenge from Phil Jagielka following his yellow card for the Barry foul.
And there were further questions for referee Lee Mason from the Everton boss after Wickham's indiscretion for the penalty was not greeted with a red card.
"You can agree or disagree with the laws but they are there," Martinez added.
"It was a penalty, a clear-cut opportunity and a red card. We had the momentum and it could have been very different.
"I was surprised Wickham was not sent off. It is a clear decision. There is no intention whatsoever – it is just a striker trying to win the ball in the box and being late. Seamus makes a fantastic run and he gets the wrong side of Wickham. Once you give a penalty away, it is a red card. I don't agree with that rule, but that is not the debate.
"I don't want to see players sent off, but again those are the rules. Jordi had a yellow card and was then caught simulating. The referee saw it. He went to blow the whistle and decided against it. They are big moments.
"Sometimes they go in your favour, sometimes they don't. Two big calls didn't go our way."
