The visitors were twice inches away from taking the lead in the first half of the New Year's Day Barclays Premier League contest as Kevin Mirallas struck first the crossbar and then the post.
Having survived those scares, Stoke subsequently went ahead in the 49th minute through Oussama Assaidi's shot.
But Everton piled on the pressure again towards the end of the game and after Potters substitute Jermaine Pennant upended his counterpart Leon Osman in the box in stoppage time, the resulting penalty was swept in by Leighton Baines.
Martinez, whose side have slipped from fourth to fifth in the table, said after the match: "I thought today - given the conditions in terms of the wet weather and this moment in the busy Christmas period, with the squads getting really stretched - that whoever scored the first goal, that was going to be final.
"And I felt in the first half, when we hit the woodwork twice, that it was not going to be our day.
"Probably with the first real chance that Stoke had, a good strike from outside the box ended up in the back of the net and it seemed that it was the worst possible case for us at that moment.
"I don't think we were at our best by any stretch of the imagination.
"But what we showed was an incredible reaction after we conceded the goal.
"We forced Stoke back in their own box. We had two balls cleared off the line and then the penalty, earned through the magic feet of Leon Osman, was a good reward.
"Even though there is a feeling we have dropped two points, I couldn't be happier with the reaction and how we finished the game."