Robinson made two brilliant saves during the shoot-out, but it was a headed goal by the Leeds shot-stopper in the dying seconds of normal time that saved Leeds from a humiliating defeat.
Only 15 seconds remained of the four minutes added on by the referee, Mark Clattenburg, when Robinson raced forward as Leeds, trailing 2-1 to an excellent performance from the Second Division side, won a couple of quick corners.
Ian Harte took the second one short to Michael Bridges who, playing for the first time in the senior side since last December, launched a cross towards the near post.
Robinson leaped highest unmarked to power the ball past the Swindon substitute goalkeeper, Rhys Evans. His first senior goal came only days after he had a brilliant penalty save cancelled out by a dubious decision against Birmingham.
It was bad luck on Swindon, who had done enough to go through against a changed Leeds side, without Mark Viduka, Jody Morris and Lamine Sakho, and who were forced to play the whole of extra time with 10 men after losing their goalkeeper, Bart Griemink, to a red card two minutes from normal time.
Andy King's team, though defending deep back in their own half for long periods, were worthy of enormous credit for restricting Leeds - who gave David Batty his first start for 18 months - largely to half-chances, and they went ahead with a stunning free-kick a minute from the end of the first half.
From fully 30 yards out the Swindon captain, Andy Gurney, beat Robinson with a terrific curling strike. It had to be special to beat him tonight!
Griemink made a brilliant stop with his foot to deny the Leeds teenager Aaron Lennon an instant equaliser. The keeper was unfortunate not to finish the match. Although the punishment for his second offence, a foul on Alan Smith just outside the area, was fair, his first yellow card for taking the ball out of his area early in the second half looked a poor decision after TV highlights proved as much.
Swindon extended their lead with 16 minutes left, their leading scorer, Sam Parkin, neatly skipping between two defenders to turn in Sammy Igoe's low cross.
Harte handed Leeds belated hope with an almost instant reply, shooting home Lennon's corner, but it seemed history would repeat itself for King - once Reid's chief scout at Sunderland - who was in charge of Mansfield when the Third Division side knocked Leeds out of the Coca-Cola Cup in 1994.
Swindon led the shoot-out 3-2 after Harte's kick had hit the outside of a post but Roque Junior and Lucas Radebe edged Leeds ahead as Robinson first saved from Stefani Miglioranzi and then Gurney to leave The Robins rocking back to Wiltshire wondering how they had failed to progress.
But this is what cup games are all about!
Leeds United (4-4-2): Robinson; Kelly (Radebe, 37), Camara, Roque Junior, Harte; Lennon, Batty, Olembe, Wilcox (Domi, h-t); A Smith, Chapuis (Bridges, 60).
Subs not used: Carson (gk), Johnson.
Swindon Town (5-3-2): Griemink; G Smith (Herring, 116), Heywood, Gurney, Reeves, Duke; Igoe (Evans, 88), Hewlett, Miglioranzi; Stevenson (Howard, 63), Parkin .
Subs not used: Viveash, Nicholas.
Referee: M Clattenburg (Co Durham). Attendance: 29,211.
Man-of-the-Match: Paul Robinson.