Oussama Assaidi put the hosts ahead four minutes into the second half, before Leighton Baines levelled from the penalty spot late in the game after Jermaine Pennant recklessly hacked down Leon Osman.
And Martinez feared his side were going to open 2014 with a defeat, but felt they ultimately got what they deserved, with Kevin Mirallas having rattled the woodwork twice in the first half.
"I thought whoever was going to score the first goal, that was going to be final," he said. "You feel in the first half when we hit the crossbar twice [post once, bar once] that it's not going to be our day.
"The first proper chance that Stoke has - a good strike from outside the box - ends up in the back of the net.
"I don't think we were at our best by any stretch of the imagination, but what we showed is an incredible, incredible reaction. Once we conceded the goal, we forced Stoke back in their own box.
"The penalty through the magic feet of Leon Osman was a good reward.
"I couldn't be happier with the reaction that we had and in the manner that we finished the game."
Everton sported an unfamiliar centre-back partnership of Antolin Alcaraz and John Stones, with Phil Jagielka sidelined and Sylvain Distin ruled out for 10-12 days with a "little nip in his hamstring".
And Martinez feels his backline has been left in safe hands.
"I couldn't be happier with the partnership," he added. "I thought Antolin was magnificent, and then you've got young John Stones who I think is going to be one of the best defenders in the English game.
"He's a born leader and I think we're going to see something special in John Stones' career.
"John Stones has got everything to be a good, modern centre-half."