Former Hibs midfielder John Rankin became the first player to miss in the shoot-out with the 13th kick, but Radoslaw Cierzniak then saved the next two penalties from Matthew Kennedy and David Gray to secure United’s passage to the last four.
The visitors led three times with goals from Chris Erskine, Aidan Connolly and Ryan Dow, with Hibs responding through Dominique Malonga, Jason Cummings and Kennedy.
United struck after 12 minutes when Conor Townsend played the ball inside to Connolly and he knocked it on to Erskine who took one touch past Liam Fontaine and kept his composure to tuck his shot past the keeper.
Hibernian were level five minutes later as the United defence were opened up by a wonderful pass from Scott Robertson and Malonga crashed his angled drive into the far corner.
Just two minutes later United restored their advantage after Rankin slipped a pass through for Dow, who may have been clipped by keeper Mark Oxley but he scrambled back onto his feet to set up Connolly in front of goal, although Lewis Stevenson
Hibernian were back on level terms 12 minutes after the break when Stevenson’s teasing cross was headed into the near corner of the United net by Cummings.
Parity only lasted four minutes this time before Dow dipped down to head the ball into the corner from eight yards.
But Hibs refused to buckle and hit back yet again as substitute Kennedy (78) drifted in from the left wing and his shot loops off the back of a United defender, over Cierzniak.
The keeper would have the final say though with two penalty saves in the shoot-out, either side of Townsend’s successful spot-kick, to clinch a semi-final place for United.