Chairman John Bennett's resignation on health grounds on Saturday came on the back of stinging criticism for the board and team in the wake of a 3-0 defeat to Celtic two weeks ago.
Philippe Clement's men began the day eight points adrift of their Old Firm rivals and Aberdeen at the top of the table.
But they got the three points they badly needed thanks to Tom Lawrence's clever dinked finish after just seven minutes.
United had been unbeaten in their opening four league games of the new season but rarely threatened to strike back as they failed to register a single shot on target.
Cyriel Dessers should have doubled Rangers' advantage before half-time when he fired straight at Jack Walton.
However, Rangers were not made to pay as they held out in the second half to climb above United into third.
