The Latics - defending the trophy they won so memorably at Wembley against Manchester City last May - started strongly and were deservedly two goals up inside 25 minutes.
Roger Espinoza fired the opener on 18 minutes with a superb strike from fully 30 yards that flew into the top corner and when Jordi Gomez made it 2-0 shortly after it seemed not a matter of if but by how many.
However, Wigan failed to put the game to bed and were pegged back with two goals in the space of two minutes just before the interval.
Reeves then repeated the feat by steering a cross from the left into the same corner with Al Habsi again helpless.
Shaun Williams' connection was true enough, but Al Habsi was across to his right like a flash and made a superb diving stop.
Wigan looked to have made the most of their fortune when they regained the lead within seconds. Callum McManaman sprung the offside trap and composed himself before smashing past David Martin.
But the Dons were not finished and, with eight minutes remaining, Patrick Bamford fired home from close range to the delight of the away supporters behind the goal.
The visitors probably deserve another crack for the way they dug in, but Wigan will be wondering how they failed to win a game they bossed for the vast majority.
James McArthur and Grant Holt had decent chances before the home side did break the deadlock, with McManaman and McClean also blowing a two-on-one break when a goal looked inevitable.
Then, leading 3-2 inside the final quarter, substitute Marc-Antoine Fortune raced into the penalty area with McClean unmarked to his left. A simple ball across would have booked Wigan's place in round four, but the striker elected to go himself and pulled the ball horribly wide to the disgust of his colleague.
That miss was compounded when Bamford equalised in the closing stages, and Wigan were the side hanging on during the five minutes of stoppage time.