Callum Paterson opened the scoring for the visitors with a 21st-minute goal, cancelled out by Johnny Russell on 35 minutes.
After Paterson hit the post, Jambos midfielder Darren Barr was sent off in the 78th minute for a second bookable offence.
United went closest to an extra-time winner when John Rankin hit the post in the 119th minute before the disjointed affair went to a shootout.
The teams remained level after the regulation five penalties apiece and after Hearts captain Marius Zaliukas converted the 13th spot-kick, Sean Dillon stepped up to fire wide.
Hearts join Inverness, Celtic and St Mirren in the semi-final draw on the afternoon of Thursday, November 8.