After two successive league defeats, it was a greatly improved performance by Notts, so much so that it became their best yet achieved under caretaker player-manager Ian Richardson.
Notts showed more commitment and greater passion with nobody doing better in the early stages than young striker Shaun Harrad on his first appearance of the season.
Causing endless problems to Swindon's back three, Harrad was unlucky not to score in the 25th minute when a lovely overhead flick beat Rhys Evans before cannoning off the goalkeeper's right-hand upright.
But within five minutes Notts took a lead they had long threatened. When the ball was played through the middle, Gavin Gordon latched on to it and, shooting on the run, he saw his shot take a deflection to leave Evans stranded.
It had not been all one-way traffic, although Notts dominated the opening 45 minutes and should have been more than one ahead at the break. The slenderness of the lead gave visiting manager Andy King a chance to fire up his team for a second-half comeback.
But Notts floored them by hitting a second goal with just over one minute of the new period played.
A corner, intelligently played, resulted in Tony Scully's cross to the back post finding Gordon, who leapt high to nod his and County's second.
Swindon then produced their most sustained period of pressure in the entire game but the Notts back three in Richardson's preferred 3-5-2 formation proved unbeatable.
Kelvin Wilson - being watched by many bigger clubs - and veteran Mike Whitlow held firm but top billing went to David Pipe whose work as sweeper was outstanding.
It was for hacking down Pipe on the edge of the penalty area that Swindon substitute Grant Smith saw a red card late on.
Notts failed to capitalise on other scoring opportunities, Matt Williams, Scully and Chris Palmer all being remiss while Palmer had another, better effort well saved by Evans.
The save was matched at the other end when Saul Deeney turned a Sam Parkin effort round the post.
The result leaves Swindon without an away win in the FA Cup since 1992.