Swindon went into the game with a one-point advantage over Karl Robinson's side but the visitors produced a fine second-half display to take the points.
Daniel Powell opened the scoring on the hour mark, heading home on the rebound after Lee Hodson's shot had been saved by Wes Foderingham in the Swindon goal.
A curling Will Grigg effort doubled the lead before the striker added his second two minutes from time.
MK Dons are now five points behind second-placed Preston North End in the race for automatic promotion to the Championship.
Meanwhile, Sheffield United tightened their grip on a play-off place by beating Barnsley 2-0 in the South Yorkshire derby.
Steve Davies gave United the lead in the 42nd minute with a header into the corner from Kieron Freeman's cross.
And the win, which gives Nigel Clough's men a seven-point gap to seventh-placed Bradford City, was wrapped up when Jason Holt turned home in the 71st minute.
