The first goal came courtesy of great wing play by Jack Grealish, who teed-up Alan Sheehan in the box and he drove a terrific low shot into the corner of the net – his sixth goal of the season.
Swindon, who had won their last three games, gradually hauled themselves back into the contest with Alex Pritchard going close.
Notts should have extended their lead when Ronan Murray raced clean through on goal but Wes Foderingham saved his effort.
The game then erupted into a mass brawl late on with Troy Archibald-Henville dismissed for an off-the-ball swipe at Jimmy Spencer, and Nathan Thompson was also sent off after collecting a second yellow.
Notts then made the game safe in injury time as Jamal Campbell-Ryce broke clear to set up Callum McGregor for a simple goal after Foderingham had gone up field to attack a free-kick.