The visitors took control of the game with strikes either side of the break.
Hearts captain Danny Wilson notched a late consolation, but it was not enough to deny United a seventh win in eight games in all competitions - a run which has helped them move within four points of third-placed Motherwell.
The result left the Edinburgh side on the brink of being condemned to the second tier for the first time in 33 years, with Gary Locke's team 21 points adrift at the bottom with only 24 left to play for.
United made an encouraging start on a drenched surface but were restricted to efforts from outside the box.
Midfielder John Rankin sent a wild drive well over Jamie MacDonald's bar from 25 yards in the seventh minute before Andrew Robertson pulled a low left-footed shot harmlessly wide.
The hosts grew into the match and enjoyed periods of sustained possession in the middle of the park.
Hearts left-back Kevin McHattie registered the first shot on target but his fierce drive from distance was straight at United goalkeeper Rado Cierzniak.
Chances were at a premium but Hearts forced Cierzniak into a low block in the 23rd minute.
Callum Paterson found Dale Carrick on the edge of the area and the striker made space with a quick turn but his low shot was pushed clear by the Polish stopper.
The visitors went ahead in the 35th minute when Dow and Robertson combined to provide Graham with a simple tap-in under the bar. Dow found Scotland full-back Robertson in the box and his teasing low cross was lifted high into the net by the marksman.
Hearts spurned a chance to level five minutes before the break when Paterson headed over with the goal gaping. David Smith's free-kick was misjudged by Cierzniak but Paterson was unable to capitalise.
Ryan Stevenson then drilled a rising effort from close range into the side-netting for Hearts after a Jamie Hamill flick in the box had created an opening.
United were quick out of the blocks in the second half and mustered two great chances within two minutes.
The first arrived in the 49th minute when Robertson raced onto a Graham pass but the full-back's stinging drive from inside the area was brilliantly palmed away by MacDonald.
Ciftci then surged into the box from the touchline after beating Wilson to Sean Dillon's clearance but the Turkish forward's low shot just missed the far post.
Stuart Armstrong should have at least tested MacDonald in the 63rd minute but he appeared to completely miss Dow's cross in his attempts to glance a header towards goal from close range.
The same two combined again moments later but this time Dow powered an Armstrong cross just wide as United continued their territorial dominance.
United finally made their possession count in the 70th minute with a brilliant goal from Ciftci. The striker was allowed to advance along the 18-yard line before unleashing a rising left-footed effort into the corner of the net past the outstretched MacDonald.
Hearts pulled one back in the 78th minute through captain Wilson, the defender heading in Sam Nicholson's corner, but United held on to claim a deserved win.