Peter Houston's side, who finished third last season, took advantage of their first of five matches - four of them in the SPL, plus Sunday's Scottish Cup clash with Motherwell - in 12 days to climb into fifth place in the table with a deserved win.
United took a two-goal half-time lead as Barry Douglas headed in a Craig Conway corner after 29 minutes and after 33 minutes Dons goalkeeper Jamie Langfield failed to hold Scotland winger Conway's swerving shot from 25 yards.
Aberdeen substitute Josh Magennis pulled a goal back after 51 minutes, but Danny Swanson struck a superb third for United nine minutes later as Houston's men moved a point ahead of Inverness and above Motherwell on goal difference with three games in hand on their rivals.
Aberdeen had gone from relegation contenders to top-half hopefuls after being revitalised since Craig Brown's appointment in December, but in a congested table remained 10th and will be reliant on other teams slipping up after finishing second best on Tayside.
Having played fewer matches than anyone else in the SPL with the split approaching, United were playing catch-up with fixtures against Hamilton on Thursday, Well in the Scottish Cup on Sunday and league games with Hearts and Inverness next week.
The Dons went close to opening the scoring when Zander Diamond, available again following an ankle injury, met Rob Milsom's corner with a firm header from six yards which Dusan Pernis saved with his legs.
Diamond required treatment on a head injury and went off, returning four minutes later with a red bandage wrapped around his head.
David Goodwillie, who performed well throughout, had three half chances before United opened the scoring with Diamond off the field undergoing further treatment for his blood injury.
Conway's corner found Douglas to head in simply at the near post, with Diamond's presence in the area missed.
The Dons defender was replaced after 31 minutes by Rory McArdle, who was wearing an identical red head bandage following a clash of heads in last week's match with Hearts.
As McArdle entered the fray, Morgaro Gomis hit the foot of the post from another corner and Langfield deflected the rebound to safety.
But soon after the Dons goalkeeper was again picking the ball out of his net as United struck for a second time in five minutes.
Conway's swerving right-foot shot from 25 yards deceived Langfield, who could only get a hand to the ball and diverted it into the bottom corner.
Magennis replaced the ineffective Sone Aluko at the interval and pulled a goal back six minutes after his introduction.
Douglas failed to deal with Ryan Jack's lofted ball into the area and Scott Vernon outmuscled him before prodding the ball across from the byline for Magennis to head in from close range.
Rob Milsom's long shot had Pernis at full stretch as the Dons went close to a quickfire second before Goodwillie hit the post with an acrobatic effort after McArdle's weak clearing header.
Goodwillie then set up United's third as Swanson beat the offside trap to latch onto his team-mate's through ball and fire an unstoppable left-foot shot into the corner.
Pernis turned a Magennis effort from 10 yards around the post with 12 minutes remaining as the Dons continued to push forward before Goodwillie went down under McArdle's challenge inside the area, only for penalty appeals to be waved away.
Magennis had a shot held by Pernis and Gomis had an effort blocked at the other end before Goodwillie was replaced by Johnny Russell as United played out time and secured a deserved win.