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.
Skacel and Gary Mackay-Steven, available for the first time since August following knee surgery, were among the substitutes, as United made one change from the side which beat St Mirren on Saturday.
Victory in Paisley ended a six-match winless run in the Clydesdale Bank Premier League which included Hearts winning 3-0 at Tannadice in September, when Paterson scored twice.
The 18-year-old started up front in a 4-4-2 formation, joined by John Sutton, the scorer of the last-gasp equaliser against Ross County on Saturday.
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 goaStevenson 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.
Hearts responded, twice going close. Cierzniak repelled a Taouil free-kick before Sutton headed down a Grainger cross for Paterson, whose acrobatic strike came off the post and the United goalkeeper was relieved to claim the rebound.
Mackay-Steven came on for Gardyne and was immediately clattered by Andy Webster, who was booked.
Barr glanced a header wide across goal from Grainger's cross and Novikovas then shot weakly at Cierzniak as he cut inside.
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.
Five minutes into the first period of extra-time, Skacel replaced Ryan to acclaim from the home and visiting support.
After Rankin had a shot saved by MacDonald, Skacel might have won it for United with five minutes remaining.
Mackay-Steven ran to the by-line and cut the ball back for Skacel to blast narrowly over.
A late corner fell for Rankin 30 yards out and his powerful shot hit the outside of the post, but the match went 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.