The Clarets were dominant, fashioning 22 shots on goal in total, with 10 on target, but went behind in the first half when Cameron Jerome planted a header into the corner.
It looked as though the visitors, who are 16th in League One, would hang on for a famous victory at Turf Moor.
However, substitute Matej Vydra equalised with the last kick of the game to take the tie into extra-time.
The scores remained level after an additional 30 minutes of play, meaning the outcome would be decided by a shootout.
Vydra, the hero in normal time, missed from the spot to give MK Dons the advantage.
But Will Norris, on his debut, saved well from Lasse Sorensen and Ben Gladwin to put the hosts back in the driving seat.
Phil Bardsley then wrapped things up with a well-taken penalty to extend the Clarets' fine run at this stage of the competition.
1. Will Norris 6
The Clarets debutant didn't have a great deal to do, other than pick the ball out of the net. Made himself big to protect his near post when Walker got in behind in the second half. Saved well from Sorensen and Gladwin in the shootout.
2. Phil Bardsley 6
Over-committed and, as a result, was left out of position for the MK Dons goal. Got stuck in on his first start for three months and put the ball on a plate for both Wood and Brady, who failed to convert. Took his penalty tremendously well to send Burnley through.
3. Erik Pieters 5
Monitored Jerome's run when MK Dons came forward, but lost the striker at the last minute when putting MK Dons ahead. Played Walker on when the visitors got in behind in the second half, but got forward well down the left hand side.
4. Ben Mee 6
Couldn't intervene when Fraser flicked on Walker's cross to set up Jerome, but proved impenetrable otherwise. Made some excellent blocks/interceptions inside the box, saw a header saved by Nicholls and almost set up Tarkowski for an equaliser.