The Clarets needed a 94th minute strike from Matej Vydra before the steadiness from the spot secured a place in the last 32.
Sean Dyche’s side were comfortably the better team in the first 45, but despite carving out chance after chance, Dons keeper Lee Nicholls proved a one-man barrier.
And with Burnley banging on the door with no luck before the break, the visitors took full advantage as Cameron Jerome nodded past Will Norris.
That setback seemed to dent Burnley’s confidence after the restart with Robbie Brady’s drive their best effort.
But despite looking set for an FA Cup giant killing, the cup gods were shine on Dyche for once, as substitute Vydra tapped in Joel Mumbongo’s free kick.
However, the Czech international almost reversed his good work in extra time as he wasted two brilliant chances and missed in the shoot out.
But Burnley keeper Norris got him out of jail with two crucial penalty stops to send Dyche’s team through.
Full Time at Turf Moor after 90 minutes it was Burnley 1 MK Dons 1 with the Clarets edging out a 4-3 win on penalties.