Having gone in front during the first half, the visitors were suddenly trailing due to errors in defence and needed Peacock to pop up again and earn them a point.
The result was of little help to either team as they bid to close the gap between mid-table and the League One play-off places.
Oldham stretched their unbeaten home run to five games, while the plus point for Swindon was a good reaction to their heavy weekend defeat against Walsall.
After a poor quality opening for the match, Swindon took the lead on 29 minutes after Christian Roberts caused problems down the left.
Peacock's initial header was well saved by Oldham goalkeeper Mark Crossley, but the striker-turned-midfielder was quick to the rebound and drove low into the net from six yards.
Swindon had further chances as Oldham struggled badly before half-time, with Billy Paynter testing Crossley and forcing another fine save.
But five minutes after half-time, Oldham pulled level through on-loan David Livermore, who got the final touch six yards out after Jason Jarrett shot from just inside the box.
Oldham's next attack saw them go in front as Mark Allott drove goalwards from 16 yards and visiting keeper Phil Smith was guilty of a bad blunder as the ball got away from him and squirmed over the line.
But there was another twist three minutes later when Peacock got the final touch from close range to bag his second from a corner, although Simon Cox also claimed the credit.
Swindon went close to a winner through Chris Allen and Paynter, while Oldham's best chance of the last half hour was blazed wide by Gary McDonald.