Brian Graham put the visitors ahead against the run of play in the first half, and Nadir Ciftci added a second on 70 minutes before Danny Wilson pulled one back to ensure a tense finale.
But United held on to move within four points of Aberdeen and Motherwell, while Hearts remain 21 points adrift of safety at the bottom as relegation looms closer.
Both sides struggled to come to terms with the wet and slick conditions in the early stages before John Rankin drilled over from distance and Andrew Robertson fired wide moments later.
But Radoslaw Cierzniak was the first goalkeeper to be tested as he got behind a decent long-range effort by Kevin McHattie, and the United keeper then did well to palm away a smart turn-and-shot from Dale Carrick.
The visitors hit back as Stuart Armstrong headed over from Ryan Dow's cross before United took the lead on 35 minutes.
Dow powered into the Hearts area and fed Armstrong, whose well-weighted cut-back teed up Graham to smash an unstoppable finish into the roof of the net.
Hearts almost equalised within a couple of minutes after Cierzniak mishandled a David Smith free-kick, but Callum Paterson missed the target with a free header inside the six-yard box.
Ryan Stevenson sliced a shot into the side-netting just before the interval, and Robertson almost doubled the lead straight after the restart as his angled strike forced an excellent one-handed save from Jamie MacDonald.
Armstrong glanced a header wide and Dow scuffed a shot from a promising position as United continued to press, and their dominance was rewarded with a second goal 20 minutes from time.
Ciftci pounced on a slack pass from Wilson and skipped past two lacklustre tackles before hammering a superb 20-yard shot beyond the reach of MacDonald.
Wilson then atoned for his error eight minutes later as he rose above the visiting defence to power home a towering header from Sam Nicholson's corner.
But United's defence stood firm over the closing stretch, and Hearts now take on Edinburgh rivals Hibernian next week knowing that they could be relegated if results go against them.