In a match where Roberto Mancini's men were far from their best, Dzeko's header midway through the first half proved to be enough to claim the victory.
Wigan pressured City's defence early on but the deadlock was broken as Dzeko brilliantly met David Silva's free-kick.
Sergio Aguero and Dzeko both had late chances to double City's lead but they were able to hold off all late attempts from a disappointing Wigan.
Wigan stay 20th in the table and will now face QPR on Saturday.
Manchester City will welcome Tottenham Hotspur on Sunday.