Reserving their best performance of the term, the tangerine teenagers triumphed at Tannadice against the Ibrox league leaders.
Both goals came in the second half when United were clearly in the ascendancy.
The first half was even, with the best chance falling to the hosts when Louis Swanson just failed to make contact with Ryan Dow's cross-cum-shot.
The goal that broke the deadlock came after 63 minutes when Gavin Stokes blasted a 25-yarder against the bar and John Murray leaped salmon-like to head home the rebound.
Fifteen minutes later, the visitors were lucky not to fall further behind, David Shaw fractionally away from getting on the end of a Scott Allan cross.
However, with twelve minutes remaining, United got the second goal they deserved.
Dow cottoned on to a quickly taken Ross McCord free-kick and though the keeper saved his first effort, Dow drove home the rebound.
Though Rangers rallied in the closing stages, but the Tannadice rearguard held firm.
Dundee United: Grant, Lee, R Smith, Howie (Irvine 39), Shaw, L Swanson (Scott 90), Allan, Stokes, Ross McCord, Dow, Murray (Thomson 84). Subs not used: Hindley, Robertson, Burns.