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."
Stoke manager Mark Hughes had no complaints about the decision to award the spot-kick for winger Pennant's tackle.
Hughes said: "He (Pennant) has made a mistake - he has made a challenge in the box and in those situations you want full-backs or defensive-minded players to be in those positions.
"Unfortunately he miscalculated and it was a stonewall penalty.
"He is an attack-minded player. It is not one of his strengths, trying to stop balls into the box and opposing attacking players."
The 12th-placed Potters went into the game against Everton, who beat them 4-0 in the reverse fixture on November 30, having lost their last two matches of 2013 by similarly comprehensive scorelines - 5-1 at Newcastle and 3-0 at Tottenham.
And Hughes added: "We have taken one point off a very good Everton team and maybe before the game started we would have been happy with that.
"I thought in the first half we played really well, and then in the second half we came out and got a good goal from Assaidi (who joined Stoke on loan from Liverpool earlier this season and has now netted four times for the Potters). He is really playing well at the moment and showing the threat he has.
"You know at some point Everton will throw people forward, take chances and try to make something happen, but I was really happy with how we showed what was required, putting our bodies on the line and blocking shots.
"You get to within a couple of minutes of the end of the game and you expect to see it out, but we made a mistake and we ended up with only one point.
"But only a few weeks ago, we were getting beaten quite comprehensively by the same team at Goodison Park, so we turned that around.
"And we have obviously had a difficult time over Christmas in terms of travelling and those results, but we have picked ourselves up today and put in a really good performance in my view."
Hughes confirmed Potters goalkeeper Thomas Sorensen had come off at half-time due to an Achilles problem.
Stoke already have first-choice stopper Asmir Begovic out injured with a broken finger, so Jack Butland came on for his debut.
Meanwhile, Martinez has said Everton defender Sylvain Distin, who did not feature at the Britannia Stadium because of a hamstring complaint, is likely to be sidelined for 10 to 12 days.