With Celtic's game postponed earlier on Saturday, victory for the Scottish champions would have seen them leap-frog their rivals to the summit.
But Rangers were twice pegged back when goals from Nikica Jelavic and Steven Naismith were cancelled out by David Robertson and Johnny Russell before David Goodwillie netted a last-minute winner.
Rangers were forced to make a number of changes from their Co-operative Insurance Cup winning team for the visit of United.
Goalkeeper Allan McGregor was back between the sticks as Neil Alexander settled for a place on the bench, after helping secure the first trophy of the season at Hampden.
Richard Foster and El-Hadji Diouf also started at Ibrox but, with the likes of Madjid Bougherra, Kyle Lafferty and Vladimir Weiss among those injured, Walter Smith's selection problems were highlighted by the fact he was able to name just five substitutes.
United also made changes from their midweek Scottish Cup defeat, with Morgaro Gomis, Barry Douglas and Russell all drafted into the side.
"Step up and Pay, Mr Whyte" urged a banner in the crowd as the Scottish tycoon's protracted takeover of Rangers neared its conclusion.
But the players were keen to focus on business on the park in their bid to move top of the table and they enjoyed much of the early possession.
Diouf's cross from the left found Jelavic in the box and the Croatian striker tried to nod past goalkeeper Dusan Pernis for the opener but his header bounced wide.
However, Jelavic was on target as Rangers surged into the lead with 19 minutes gone.
He won a free-kick just outside the area before connecting with Diouf's ball into the box to send a firm header into the back of the net from 10 yards.
United had tried their luck with a couple of pot shots from outside the box that had failed to trouble McGregor.
But they were back on level terms a minute from half-time when David Robertson raced onto Danny Swanson's pass and slotted home.
Rangers knew they needed a positive reaction in the second half and they managed to restore their lead eight minutes after the restart.
Diouf's corner broke for Maurice Edu and he nodded on for Naismith to rifle past the helpless Pernis high into the back of the net.
United must have thought they had levelled again when Swanson fired through the legs of McGregor from a tight angle only for the ball to trundle across the face of the empty goal before the danger was eventually cleared.
At the other end, another corner from Diouf fell for Naismith but, this time, he blasted well over, squandering the chance to double his tally.
United refused to give up and had the net bulging again after 78 minutes, with Paul Dixon setting up Russell for a header that left McGregor with no chance.
The Rangers goalkeeper was then called into action to tip a Goodwillie effort onto the bar as the visitors attempted to go for the win.
With time fast running out, Jelavic threw himself in front of Gregg Wylde's cross only to see his header blocked, before Steven Whittaker saw his header come off the bar.
Instead, it was United who snatched all three points as Goodwillie immediately raced up the other end of the park and fired home the winner with a minute to go.