Striker Jon Daly struck with a header from Barry Douglas' free-kick after 41 minutes and defender Gavin Gunning headed in a second from a corner after 85 minutes as United triumphed to move up to fifth place, two points behind St Johnstone.
Hearts began the day in fourth but now have one win in seven matches - the extra-time defeat of St Johnstone in the Scottish Cup - and sit sixth in the table.
The hosts' woes were compounded by injuries to midfielder Ian Black and captain Marius Zaliukas and they are just four points ahead of seventh-placed Aberdeen with the SPL split approaching.
It is a run of results which may bring into question Paulo Sergio's position as Hearts manager.
Sergio made five changes following the 3-0 loss at Motherwell, but new signing Craig Beattie was not even among the substitutes.
Jamie Hamill, Darren Barr, Rudi Skacel, Mehdi Taouil and Scott Robinson returned to the starting line-up, while United made one enforced change following their 4-0 defeat of Kilmarnock, with Barry Douglas replacing the suspended Paul Dixon at left-back.
The hosts began brightly with Skacel and Robinson each forcing Dusan Pernis to save low to his right.
Hearts should have gone ahead when an unmarked David Obua headed Hamill's cross down to Stephen Elliott, who was allowed to turn and shoot from 10 yards, but Pernis saved again.
United were playing on the counter attack and Barr was booked for bringing down Johnny Russell before Jamie MacDonald palmed over a Sean Dillon header from Willo Flood's corner.
Pernis saved a dipping Black volley from the edge of the area before Russell blasted off target at the other end.
The physical nature of the match saw Barr perhaps fortunate to avoid a second booking for a tackle from behind on Daly and his subsequent reaction.
Black, who required treatment earlier following an aerial challenge with Daly, went off after 40 minutes following another physical challenge and was replaced by Ryan McGowan.
Moments later Hearts suffered a further blow when Douglas' free-kick from wide on the left was whipped to the far post and Daly stooped low to head beyond MacDonald. It was his 17th goal of the season.
Robbie Neilson cleared off the line as Hearts went in search of an equaliser before the interval.
Zaliukas did not reappear after the break following a first-half injury and the captain's departure led to a reshuffle, with Suso Santana coming on and McGowan dropping back into defence.
Hearts left-back Danny Grainger went to ground too easily in the area after 50 minutes with four United players nearby, but referee Calum Murray dismissed appeals for a penalty.
Flood was booked for a late tackle on Grainger soon after before United replaced Russell with Stuart Armstrong.
A Neilson lunge on Grainger saw the former Hearts defender yellow carded as both sides struggled to assert their authority.
Hearts twice came close to an equaliser in quick succession after 79 minutes.
Santana threaded a neat ball through to Skacel, but Pernis snuffed out the shot by swiftly breaking off his line and spreading himself.
The ball came back for Elliott, but Gunning blocked the shot, with Pernis also scrambling to cover.
United responded and were rewarded with a second goal.
A Gary MacKay-Steven corner after 85 minutes was met by Gunning, who headed into the top corner.
Gunning headed wide moments later from another set-piece and United held on to secure victory.