Yanic Wildschut went close early on for the Latics when he cut inside and drove a shot against the upright.
Billy Clarke failed to make contact with Jamie Proctor's dangerous cross for the visitors after the interval.
And Vuckic ensured Wigan would remain second in the table with a composed finish before Wildschut drove wide as he attempted to add a second.
Defeat cost play-off hopefuls Bradford their place in the top six, with Yorkshire rivals Barnsley winning at Fleetwood.
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Wigan manager Gary Caldwell told BBC Radio Manchester: "I think we were always the team on the front foot. They made it difficult for us.
"I thought we kept going and showed great character, great belief in what we were doing, kept passing the ball and got our reward in the end."
Wigan extended their unbeaten run to 16 games as Haris Vuckic's goal gave them victory over Bradford City.