Woodman – who spent the second half of the 2016/17 campaign on loan at Rugby Park – kept out spot kicks from Eamonn Brophy, Greg Taylor and Greg Kiltie after the two sides had drawn 1-1 after 120 minutes.
Stephen O’Donnell opened the scoring for Killie six minutes into extra time, slotting beyond Woodman after the home side countered quickly.
However, Kenny McLean levelled from the spot after Stevie May was fouled by Kirk Broadfoot, the central defensive partner of another Magpies loanee, Stuart Findlay.
That meant that the lottery of a shootout would be necessary to determine who would go on to face Motherwell in the last four. Both teams got off to a successful start, before Woodman superbly tipped Brophy’s effort onto the woodwork to hand the Dons the advantage.
The England Under-21 international then denied Taylor with his feet, though Niall McGinn’s miss for Aberdeen gave O’Donnell the chance to restore parity, which he duly did.
However, Greg Stewart smashed home to put the visitors 3-2 up, before Woodman got down low to his left to push Kiltie’s attempt away and secure a memorable win for Derek McInnes’ side.