The administration-hit and makeshift Gers, weakened by injury and suspension, found themselves two down following United defender Keith Watson's wonderful first-half drive and a Jon Daly goal soon after the break.
A deflected strike by Rangers substitute Sone Aluko, on for debutant Andrew Mitchell, off John Rankin, brought the visitors back into the game but it proved to be a mere consolation for a club whose season is in danger of becoming traumatic on as well as off the park.
While the Ibrox club continue to search for a new owner, Ally McCoist has to somehow see out the season in a positive mode but his task was made no easier with six defenders missing for the trip to Tayside.
Carlos Bocanegra and Dorin Goian were suspended, Steven Whittaker (hernia), Kyle Bartley and Kirk Broadfoot (both knee) are still sidelined and Sasa Papac had picked up a knock leaving the visitors unable to fill the bench.
Midfielder Rhys McCabe, making only his second appearance, dropped back to play right-back with Lee McCulloch moving from centre forward to centre-half beside Ross Perry.
Maurice Edu returned from suspension as Northern Ireland Under-21 player Mitchell made his debut with a rare start for Alejandro Bedoya, his last appearance being last December against Inverness.
In contrast, United only had right-back Robbie Neilson suspended, his place taken by Watson who was to prove the difference in the first half but not before Rangers had put up a spirited show.
The Govan men looked disjointed in the early stages with Andrew Little playing the lone striker's role but there was plenty of tenacious enthusiasm.
As confidence grew slowly, skipper Steven Davis tested Tannadice keeper Dusan Pernis in the 12th minute with a fine left-footed drive, after taking a Little pass but the home side defended the corner.
The match became tetchy for a spell and Mitchell picked up a booking for persistent fouling, after Watson had been cautioned for a hefty challenge on Ibrox left-back Lee Wallace.
United began to work their way into the game but in the 28th minute dithering defender Sean Dillon was robbed by Salim Kerkar, with the ball breaking to Davis.
However, the Gers wide-man was too slow on to the return pass, allowing Pernis to rush from the edge of the box and clear.
Moments later, Bedoya headed a Wallace cross over the bar before Kerkar slalomed his way through some inept challenges in the United box and toe-poked the ball wide of the target.
It was encouraging stuff for McCoist but with their first effort on goal, the home side found themselves in the lead.
When the Rangers defence failed to clear properly, Paul Dixon's throw-in from the left was headed out by McCulloch but only to Watson at the edge of the box and he fired high past McGregor with a glorious strike.
With with less than two minutes played of the second-half, and after United striker Johnny Russell had missed the target with a drive, the home side doubled their lead.
Winger Gary Mackay-Steven raced past McCabe all too easily and when his cross came in to the box skipper Daly efficiently knocked it past McGregor from close range.
Rangers were in danger of unravelling and McGregor made a save from Daly's attempted chip at the second attempt, before Aluko replaced Mitchell.
In the 56th minute, with no sign of Rangers coming back into the game, Tannadice defender Gavin Gunning headed a Dixon free-kick over the bar.
But in the 59th minute the visitors were handed a lifeline when Aluko survived a couple of poor tackles near the by-line before his shot from a tight angle was deflected off Rankin and over Pernis.
As Rangers rallied, Bedoya's left-footed drive was easily saved by the Tannadice keeper.
McCulloch, at the other end, bravely headed Russell's goal-bound drive over the bar after Gunning had headed down a Dixon free-kick and Rangers emerged unscathed from the corner.
Rangers had regained their earlier enthusiasm although United still looked the more composed of the two teams.
In the 73rd minute a Little volley from a Davis pass went straight to Pernis before Edu appeared to lash back with his arm at Scott Robertson as he tried to hold the United midfielder off but referee Brian Winter appeared to miss the incident despite being well-placed.
In the fraught closing stages Rangers pressed valiantly for the equaliser but Pernis was never put under any real pressure, leaving United to take the points.