On-loan Tottenham midfielder Alex Pritchard's stunning first-half opener was cancelled out by Dons striker Bamford, after Nathan Byrne's 51st-minute dismissal for stamping on Shaun Williams.
Swindon carried the greater threat in the first half and Pritchard put them ahead after 17 minutes, curling a brilliant 25-yard free-kick into the top corner after Massimo Luongo had been tripped by Darren Potter.
Dons goalkeeper David Martin had to be alert to deny Ryan Mason, Byrne and most notably Nile Ranger, while Stephen Gleeson fizzed a deflected long-range shot just wide for the hosts.
Swindon were reduced to 10 men six minutes after the break when Byrne maliciously stamped on Williams' foot as the Dons defender cleared the ball.
Bamford equalised for the Dons just four minutes later, racing on to Potter's defence-splitting pass to poke the ball past Wes Foderingham for his fifth goal of the season.
The Dons pressed for a winner in the closing stages, but Danny Green's angled curler was well-saved by Foderingham before substitute George Baldock had an 89th-minute goal controversially ruled out for offside.