The Latics pulled off the shock win at the Emirates Stadium to continue their dramatic late bid to escape the Premier League relegation zone.
Martinez's side scored twice in the first eight minutes and although Arsenal struck back quickly, Wigan hung on to claim a victory that moved them five points clear of relegation with four matches left in the season.
Arsenal stayed third, five points ahead of their north London neighbours Tottenham Hotspur and fifth-placed Newcastle United.
"I'm so so satisfied because that wasn't an accident," Martinez said.
"In eight days, to have to play Chelsea, Manchester United and Arsenal is such a big task, but today in the manner that we coped physically and tactically, everyone at Wigan Athletic should be very proud.
"In the second half, to have that mentality of know-how in such a young group, to understand what's needed in different phases, we coped with an outstanding footballing side.
"It was a well-deserved three points, and to say that speaks volumes of the performance of the players."
Martinez added: "We need to make sure we carry on with the standard that we showed today and we did on Wednesday, and we make sure that we finish the season as strong as we can be because 34 points is still a small amount at this stage of the season."
Wigan, who beat league leaders Manchester United 1-0 last week, took the lead when striker Franco di Santo scored a classic counter-attacking goal after seven minutes and they immediately doubled the advantage through Jordi Gomez.
Arsenal, who had won nine of their last 10 league matches, were shell-shocked but they regrouped quickly and defender Thomas Vermaelen powered home a header after 21 minutes from Tomas Rosicky's whipped cross.
"We should have been ahead at half-time but in the second half we did not create a lot because we were not patient enough," Arsenal manager Arsene Wenger said.
"In the second half we lost our cohesion, we were doing too much individually, the passing wasn't quick enough to be dangerous. We had too many strikers, we were exposed a bit but we weren't sharp enough."
Wigan climbed three points ahead of 17th-placed Queens Park Rangers and five clear of Bolton Wanderers who occupy the last relegation spot but have two games in hand on their rivals.