Sean Dillon's sudden death penalty miss saw 10-man Hearts progress to the Scottish Communities League Cup semi-final with a 5-4 shootout win over Dundee United after the game finished 1-1 after extra time.
Callum Paterson put the visitors in front after 20 minutes and hit a post in the second half after Johnny Russell's 35th-minute equaliser. Hearts midfielder Darren Barr was sent off for a second bookable offence after 78 minutes and United went closest to an extra-time winner when John Rankin hit the post in the 119th minute before the disjointed affair went to penalties.
The teams remained level after the regulation five penalties apiece and after Hearts captain Marius Zaliukas converted the 13th spot-kick, Dillon stepped up to fire wide, sending the visitors into the last four alongside Celtic, Inverness and St Mirren.
Rudi Skacel, who left Hearts after winning the Scottish Cup in May and trained with the Tynecastle club before joining United last week, came off the bench in extra-time but was not selected to take a penalty.
Ryan Stevenson required attention to a head injury sustained in a challenge with Dillon in a scrappy opening, but the Hearts midfielder recovered to set up the opening goal.
Stevenson found space on the Hearts left after Willo Flood failed to win the ball from Arvydas Novikovas and hit a low cross which Paterson met at the near post to tap in.
Boosted by the goal, Hearts went in search of a second. Danny Grainger forced Cierzniak to save and Mehdi Taouil shot over, but then United hit back with their first real opportunity.
A Keith Watson pass found Michael Gardyne on the edge of the area and his shot was blocked, only for the ball to break for Russell to prod in beyond Jamie MacDonald.
United started the second half on the attack, with Stuart Armstrong shooting narrowly wide. But as the half settled down and just as the visitors appeared to be getting on top they were reduced to 10 men as Barr, who was booked earlier in the second half for pulling down Flood, was cautioned for a poor challenge on Russell in an innocuous position.
United were unable to make their numerical advantage pay and Hearts were content to play for the additional time, before the clash wne to penalties. Grainger and Sutton missed Hearts' first two penalties, but misses from Mackay-Steven and Rankin meant the tie went to sudden death, with Dillon's miss decisive.