Luke Chadwick's early goal for a long time looked like producing a shock for the League 1 side but the teenager on loan from Manchester United swung the match the Latics' way.
Uwe Rosler remains unbeaten in domestic matches as Wigan manager and he can now look forward to a home tie against Crystal Palace.
It was the Dons who struck first after 10 minutes as Lee Hodson's hanging cross was met with a controlled volley from Chadwick, playing his first match since November.
Fraser Fyvie then came close to an excellent equaliser for Wigan as his powerful long-range effort drew a very good save out of goalkeeper David Martin, who clawed it around the post.
But MK Dons themselves looked lively and almost doubled their lead when Wigan stopper Ali Al Habsi tipped away Shaun Williams' free kick.
The Omani then had to be at his sharpest again to keep out Dean Lewington's half-volley before Roger Espinoza curled an effort wide from outside the box for the Latics.
Wigan began the second half on the front foot with Fyvie sending another long-range shot wide before Rosler shook things up by replacing him with Powell with half an hour left.
But the Dons began to rally again with a substitute of their own, Dean Bowditch, almost inadvertently adding a second when his deflected cross was tipped over by Al Habsi.
Wigan were hanging on and their grip on this tie and the FA Cup almost slipped as Williams saw a deflected shot on the turn come back off the far post.
But the holders drew level with 11 minutes of normal time left as a terrific cross by Chris McCann was met by the head of Powell with Martin stranded to force extra-time.
It took the 19-year-old just over a minute after the restart to put Wigan ahead as his dink over Martin clipped the post on its way in after he was picked out by Jordi Gomez's through ball.
The visitors then made sure of their progression just before the end of the first period as Marc-Antoine Fortune cut inside from the right before scoring at the near post.
MK Dons refused the give in, however, and almost forced a rousing finale as Al Habsi fumbled Stephen Gleeson's low shot from 25 yards before just about grabbing the rebound.
But they were almost caught on the break again as Espinoza's cross-shot forced a smart save from Martin to prevent what would have been a flattering scoreline.