Arsenal will play either Hull City or Sheffield United in the final.
Arsenal goalkeeper Lukasz Fabianksi, selected by manager Arsene Wenger instead of regular first-choice Wojciech Szczesny, saved Wigan’s two opening penalties from Gary Caldwell and Jack Collison.
There was no way back for Wigan from there, with Arsenal’s Santi Cazorla sealing the shoot-out win when he beat former England goalkeeper Scott Carson from the spot.
Wigan, who made history last season by becoming the first side to win the FA Cup and be relegated from English football’s top flight in the same season, took the lead in normal time.
Jordi Gomez’s 63rd-minute penalty put them ahead and left Arsenal, without a major trophy since lifting the FA Cup in 2005, looking at yet another season without silverware.
But Germany defender Per Mertesacker, who had conceded the penalty, gave Arsenal a lifeline eight minutes from time when he nodded in a shot from Alex Oxlade-Chamberlain.