Depleted Rangers, who were without six defenders due to suspension and injuries, were only able to field five substitutes for the match, as youngsters Rhys McCabe and Ross Perry joined Lee McCulloch in a makeshift back four.
The Gers held the home side at bay until the 37th minute when defender Keith Watson volleyed United into the lead.
Jon Daly added a second two minutes after the break before substitute Sone Aluko gave Rangers hope when his deflected shot beat Dusan Pernis on the hour mark.
If Motherwell fail to beat Aberdeen later today it means Celtic, who play Kilmarnock at Hampden on Sunday in the League Cup final, could win the title at Ibrox next weekend. Celtic are 21 points clear of their rivals.
Ally McCoist thought his side had deserved something from the game.
"I've absolutely no complaints about the effort and commitment my players showed today," the Rangers manager said.
"I felt apart from the first 15 minutes of the second half we were really good. In the first half we had by far the better chances to score and with a little more pressure in the second half we could have got a draw."
Peter Houston said Dundee United had won without playing well.
"We didn't play at the standards at which we normally play. We didn't pass the ball as well as we can do, we gave the ball away cheaply and we panicked a bit towards the end," the United manager said.
"But the most important thing at this stage of the season is three points in our quest to try and get a top six spot."
At Tannadice, former Man City youth player Andy Mitchell made his Rangers debut and after a good run down the right sent a decent delivery into the box but American Alejandro Bedoya couldn't direct his header on target.
Andy Little then made room for his international teammate Steven Davis to let fly with a shot from distance but Pernis made the save.
Davis then sent Salim Kerkar clear but Pernis rushed for his goal to clear the danger. The Algerian winger poked the ball just wide of Pernis' post after going on a mazy run into the box.
Rangers had kept United subdued for most of the first half but it was the home side who took a 37th minute lead thanks to a sensational strike from Watson.
McCulloch could only clear a long throw-in to the edge of the box where the United defender was waiting to send a vicious volley into the top right-hand corner of the net.
United doubled their lead within 90 seconds of the restart. Gary Mackay-Stevens ghosted past two challenges before putting a dangerous delivery into the path of Daly who swept the ball past Allan McGregor.
Rangers brought on Aluko for Mitchell to join Andy Little up front and within seven minutes he had reduced the deficit.
The Nigerian international showed great skill to evade two challenges on the by-line before his cross-cum-shot took a deflection off John Rankin to loop over the head of Pernis on the hour mark.
The Gers then had McCulloch to thank as he somehow managed to head Johnny Russell's fierce point-blank strike over the bar.
Little saw his first time effort held by Pernis after Davis picked him out in the box with a viciously whipped in cross.
The Northern Ireland striker then sent a half-volley over the bar in stoppage time as Rangers failed to haul themselves level.
Source: AFP
Source: AFP