It was very harsh on Swindon who were the better side for much of the match and were reduced to ten men when goalkeeper Bart Griemink was sent off in the 87th minute for a second bookable offence.
In the shoot-out Swindon's hopes were raised when Ian Harte hit the outside of the post with his spot-kick, but Robinson pulled off brilliant saves to deny Stefani Miglioranzi and skipper Andy Gurney to end Andy King's side's dreams of giant-killing glory.
After a first minute chance from Frenchman Cyril Chapuis on his debut, Leeds lost the plot and increasingly confident Swindon surged into a 2-0 lead with just 15 minutes to go.
Gurney curled in a great free-kick from 25 yards and Sam Parkin grabbed the second when he was put clear by Sammy Igoe.
It was no more than King's side deserved. They had been the better side in the first half and looked to have withstood a siege straight after the break.
But poor marking at the far post at a corner allowed Harte to pull one back and with three minutes left goalkeeper Griemink was dismissed for a second bookable offence when he brought down Alan Smith at the edge of the area.
The time it took to replace him with Rhys Evans was vital because in the fourth minute of added time Leeds equalised.
Aaron Lennon played a last-gasp short corner to Michael Bridges who swung it into the area where goalkeeper Robinson rose above everyone to head home.
Leeds failed to make any impression in extra-time, Smith wasting their best chance when he headed Lennon's cross wide in the last minute, setting up the penalty shoot-out.