The Spanish midfielder sent Ali Al-Habsi the wrong way from the spot on 60 minutes following Jean Beausejour's clumsy foul on Theo Walcott after neat work between the England striker and Santi Cazorla.
Wigan were the better team for long spells, but wasteful crossing from the wings and poor finishing from Aroune Kone left the Latics frustrated and winless in five league games.
Injuries forced Martinez into fielding a makeshift back line, as Adrian Lopez's hamstring tear meant James McCarthy played at centre-back for the hosts.
Wigan won 2-1 at the Emirates the last time the sides met and showed no fear in the first half, but it was Arsenal who had the first real chance.
Cazorla turned James McArthur easily and found Mikel Arteta in space. The Spaniard fed Oxlade-Chamberlain on the right, who forced Ali Al-Habsi into a fine one-handed stop from 16 yards.
Kone then broke well from midfield shortly after before shooting wastefully, before Podolski drove an effort at Al-Habsi after a fine run.
Wigan should have taken the lead after 24 minutes as Di Santo put Kone clean through on goal, but with just Wojciech Szczesny to beat, the Ivory Coast striker dragged his shot wide from eight yards.
The Gunners struggled with Wigan's high-pressure tactics, while Jean Beausejour's and Ronnie Stam's deliveries were poor as wild crosses typified a first half full of energy but low on quality.
The second half began much as the first with both teams breaking forward looking for an early goal, and Al-Habsi was forced into a great stop on 48 minutes.
Oxlade-Chamberlain found space on the right wing and pulled the ball back for Walcott, whose finish was close enough for the Wigan keeper to get down to and tip away well.
However, the game turned with half an hour to go as Arsenal found the breakthrough, with the danger again coming from the right.
Walcott and Cazorla's link-up play on the wing was too intricate for Beausejour, and referee Jonathan Moss blew for a penalty as the striker was felled by the defender, with Arteta finding the bottom left-hand corner of Al-Habsi's net.
Wigan kept coming though, and Kone did well to work the ball from the left before forcing a fine low stop from Szczesny with 20 minutes remaining, while Jones could only crack an effort narrowly wide two minutes later.
Mertesacker had to be alert to deny Kone late on in the area, while McArthur dragged a shot wide late in injury time as Wenger's side saw the game out for a much-needed victory.