Ian Harte pulled a goal back on 77 minutes and after Town's keeper Griemink was sent off for two bookable offences the unthinkable happened as Robinson came up for a corner and equalised to take the game into extra-time.
Leeds did not take advantage of their man advantage during this period but Alan Smith had a golden opportunity to seal victory in the final minute of extra-time but it was squandered.
Going to a penalty shoot-out and the drama continued with a strange catalogue of penalties, which included two from Swindon that hit the post and went in off Robinson.
Penalty Clockwatch;
2224: Leeds convert their first penalty from Alan Smith 1-0.
2225: Sam Parkin strikes the post and then Robinson before ending in the net 1-1. Lucky one!
2226: Regular penalty-king Ian Harte misses for Leeds, putting his kick wide to the right 1-1.
2227: Swindon's second penalty from Brian Howard also goes in off the post and Robinson 1-2. Another lucky one!
2228: Michael Bridges scores for Leeds, excellently taken, giving substitute goalkeeper Rhys Evans no chance.
2229:Ian Herring scores for Swindon 2-3.
2229: Roque Junior takes a superb penalty to level matters but The Robins still have a penalty in hand 3-3.
2230: Hero of the night Robinson saves from Stefani Miglioranzi 3-3.
2230: Radebe scores and The Whites take charge 4-3.
2231: First scorer of the night Gurney hits the post and Leeds have won 4-3 on penalties. What a night!