The Dons keeper magnificently turned Grant Smith's spot-kick round the post after Dean Lewington was adjudged to have handled Sam Igoe's cross in the box.
But Swindon were left to rue a host of missed chances after Smith's first-half header had cancelled out Wade Small's superb individual opener for the Dons.
Milton Keynes applied all the early pressure but it was the Robins who carved out the first real chance, when Igoe's fierce strike across the face of goal was almost turned in by the stretching Sam Parkin.
Andy King's side paid dearly for that miss when Small capitalised on some poor defending by David Duke to give the Dons a 14th minute lead.
The pacy wide-man skinned the exposed Duke on the right-hand side, strode into the box and fired home his11th goal of the season into the far corner of Rhys Evans' net.
Gareth Edds went close to extending the home side's lead with a long-range effort from a well-worked corner, before Ben Harding's fizzing drive was well-held by Evans in the Robins goal.
At the other end, Christian Roberts came within a whisker of equalising when he tricked his way past Leon Crooks but curled his shot into the side-netting.
However, Roberts got it right two minutes before the break when he cleverly carved out Swindon's equaliser for Smith. The forward found space on the right before delivering a delicious ball into the box for the midfielder to head past the despairing dive of Baker into the far corner of the net.
Parkin almost put the resurgent visitors ahead moments after the restart when he headed Igoe's cross just over the bar, and Roberts then blazed over when well-placed after Dons' veteran skipper Steve Palmer was slow to react.
Parkin wasted another opportunity as Town continued to dominate, and Smith was unfortunate not to grab his second of the game when his fierce volley from Igoe's cross hit the angle before his late penalty miss.