The Latics - who remain in the Barclays Premier League relegation zone - looked set to leave the Britannia Stadium empty-handed after a tight contest had swung dramatically in the hosts' favour late on.
Jonathan Walters scored from the spot with 13 minutes remaining, after Wigan captain Gary Caldwell had been sent off for handball, to cancel out Victor Moses' opener on the stroke of half-time.
Substitute Cameron Jerome - whose goalbound header had led to Caldwell's dismissal - then put Stoke in front with his first league goal for the club, but Wigan hit back with a penalty of their own, Ben Watson coming off the bench to send Thomas Sorensen the wrong way after Ryan Shotton had been penalised for a tug on Hugo Rodallega's shirt.
Asked whether he had taken a gamble by bringing on Watson to take such a crucial spot-kick, Martinez replied: "In football, it's not (a case of) being brave, it's trying to do the right things.
"Taking a penalty is something of a specialist (task) and we have got two specialists at the club; Jordi Gomez and Ben Watson.
"Jordi Gomez wasn't on the pitch at that time and it would have been very unfair, or a big gamble, to put another player in that position."
Watson had missed his previous two penalties, but made no mistake on this occasion.
"Only the players who take the penalties can miss," Martinez continued. "The important thing is you get a specialist in that situation.
"Ben Watson, for me, is technically one of the best players in English football and that's his forte; his concentration. The big thing is he showed that he was ready for the team. I had a real belief and confidence in him."
Martinez felt a draw was the least Wigan deserved after they frustrated their hosts for much of the game.
"Looking at the whole performance, I am disappointed just getting one point, but today [Saturday] we had to get that point through character, desire, and that belief between the group," he said.
"I felt that we defended really, really well. The biggest thing is to be able to defend against a Stoke side, we all know that.
"On top of that I thought we were a real threat on the counter, the goal from Victor Moses from open play, which is the hardest thing in football, was delightful and we had one or two other opportunities."
While the Wigan boss believed referee Michael Oliver had "got the two big calls right", Stoke assistant manager Dave Kemp was dismayed by the penalty decision that went against his team.
"I've only looked at it again briefly, I suppose you would argue there was some kind of tug that no one else managed to see," he said.
"It's not a penalty. For me, when penalties are given you expect them to be clear-cut.
"It was one of those phantom ones and we were on the wrong end of it."
Kemp conceded Stoke - who end 2011 eighth in the table - were not at their best, adding: "I was a bit disappointed with some of our quality in their penalty box.
"If we had have won the game 2-1 without the penalty, we would all be sitting here quite happy with it, but obviously that's changed now and we are disappointed not to get the three points."