United were second best for much of the first half, but took the lead against the run of play through Keith Watson's emphatic volley just before the break.
Captain Jon Daly doubled Dundee United's lead early in the second half with a tap-in as an under-strength Rangers struggled to make an impression.
Sone Aluko pulled one back for the visitors with a deflected effort after a strong run, but it proved only to be a consolation as United held on for their first league victory over Rangers since 2007.
The result means that, depending on other results, Celtic could clinch the title at Ibrox when the two sides meet in the Old Firm derby next week.
Rangers may be close to securing a new buyer but the turbulence off the pitch restricted manager Ally McCoist to name just five substitutes.
But Rangers were the better side for much of a tame first half, with Steven Davis coming closest when he was allowed to drive into space on the edge of the box and draw a save from Pernis.
But the visitors were undone just before the break when Lee McCulloch's weak header dropped to Watson, who lashed in a stunning shot on the half-volley.
Rangers' misery was compounded moments after the break when an incisive United move saw Gary Mackay-Steven deliver a perfect low cross from the left, that Daly steered in from close-range.
United had chances to add to their lead - a lead that slightly flattered them - but were pegged back when Aluko scored a few minutes after coming on.
A determined run down the right saw the Nigerian beat three United players, before his shot deflected off the knee of John Rankin and looped into the net.
It was a fortunate finish, but a goal that Aluko - and Rangers - deserved.
The goal lifted the visitors but, despite chances for Lee Wallace, Alejandro Bedoya and Andy Little late on, Dundee United defended resolutely and held on for the win.