Hearts, ravaged by injuries put out a patchwork team and United capitalised as skipper Barry Robson grabbed a glory hat-trick while Lee Wilkie put up the shutters at the back.
The windy conditions made it difficult for both sides and quality football was at a premium in the first half.
But United almost took the lead after 26 minutes when Christian Kalvenes' long ball came off Christophe Berra and Noel Hunt reacted quickly to flick the ball off the post.
And the game erupted in 33 minutes when Ibrahim Tall and David Robertson went face to face and referee Craig Mackay gave them both straight red cards. It seemed harsh but was in keeping with an inconsistent display from the referee.
The visitors adapted to being a man short much better than Hearts and went in front five minutes after the break.
Derek Stillie had just made a great save from a Linas Pilibaitis volley when Steven Robb charged 40 yards and picked out Robson's run allowing United's captain to fire a low finish past Craig Gordon.
And Robson got the freedom of Tynecastle to run on to Jon Daly's lay-off and nutmeg Gordon to add United's second on the hour.
Hearts' embarrassment was complete when Hunt got in on the act sliding home Robb's cut-back, before Robson got time and space to cut in off the left and fire in a low drive for United's fourth.