Mikel Arteta’s penalty secured a third consecutive Premier League victory for Arsenal for the first time since March.
Arouna Kone wasted the Latics’ best chance when he broke clear but failed to hit the target in the first-half.
Arteta scored from the spot just before the hour after Jean Beausejour was penalized for a challenge on Theo Walcott, who again played as a striker.
Wigan have lost six of their last eight games and are in the bottom three at Christmas for the third straight year.
Gunners boss Arsene Wenger joked about coming through a “super crisis” after the 5-2 win at Reading on Monday.
And the Frenchman will take a large measure of satisfaction with the way his team, who were unspectacular but efficient, dealt with a difficult afternoon at the DW Stadium.
With festive matches against Newcastle and Southampton to come, Wenger will be confident that his team can enter the New Year with a tight grip on a top-four spot.
Although he earned the penalty, Walcott, whose future remains uncertain, was deprived of service for much of the game and the closest he came to scoring himself was a second-half free-kick.
With Antonio Lopez out for 10 weeks, he will be desperate for Gary Caldwell, Ivan Ramis and Antolin Alcaraz to return to action over the festive period.
Martinez did cause Arsenal uncertainty when James McCarthy lined up as a third centre-back at the kick off only to take up his usual position in midfield seconds later.
But Arsenal began well and after Arteta was allowed to carry the ball from halfway, he fed Alex Oxlade-Chamberlain, whose drive was pushed past the post by Ali Al Habsi.
For all of their selection issues, Wigan played with purpose and Kone should have done better when he beat Jack Wilshere and Thomas Vermaelen before hitting a tame shot straight at Wojciech Szczesny.
The Ivory Coast striker threatened again when he was played through by Franco Di Santo and tore away from Per Mertesacker but his shot squirmed wide.
Wigan full-backs Ronnie Stam and Jean Beausejour were still getting forward at every opportunity despite only having two centre-backs behind them.
And after the interval, Arsenal began to exploit the space behind Beausejour, with Oxlade-Chamberlain having one cross cut out and then picking out Walcott, who was denied from close-range by Al Habsi.
It was a policy that paid off just before the hour when Walcott dropped wide and swapped passes with Santi Cazorla before going down under Beausejour’s challenge.
Arteta waited for Al Habsi to dive and stroked in his third successful penalty of the season.
Wigan responded with Kone cutting in from the left and forcing a save from Szczesny and Maynor Figueroa had a drive held by the goalkeeper.
David Jones thumped a shot narrowly wide from just outside the area but Arsenal came through a nervous finish to take the points.
Wigan boss Roberto Martinez: “That first goal was going to be crucial. The penalty was very soft. Theo was very clever.
“At the other end, it was a clear penalty [for a handball against Kieran Gibbs] because the hands were not in a natural position.
“We are going through a spell where the calls are not going for us, but I am proud of the players. Once all the players are back we will get the rewards.”
Arsenal manager Arsene Wenger: “It was hard-fought. Wigan is a good side. We were not as good, sharp and fluent as Monday.
“It was a tough game and Wigan can surprise a lot of teams. Once we were 1-0 up it was all Wigan but we were well organized. It looked like from outside [that it was a penalty for us].”
Asked if they can play up front, he added: “I am convinced he can do that. Today he didn’t get that much service but he worked hard and got us the penalty.”