Ally McCoist's men had already exited the Champions League, the Europa League and the Scottish Communities League Cup, meaning the SPL title is now their only hope of silverware this season.
McCoist was the last Rangers player to score a winner against United in the Scottish Cup back in 1989.
And lack of resources up front following the deadline day departure of Nikica Jelavic was painfully evident as they ended this tie with a front pairing of Salim Kerkar and Andrew Little - one player who has never started a game for the club and another who has been injured for the best part of the last 18 months.
In front of a crowd of just 17,822, Rangers made one change from last week's win over Hibernian: Mervan Celik, the club's only January recruit, was handed his first start in the fifth-round clash ahead of Gregg Wylde, who was named on the bench.
United also made a single change from last week's draw against Motherwell, with Gunning in for Sean Dillon and new signing Milos Lacny among the substitutes.
Johnny Russell had the ball in the net for United after a couple of minutes but the effort was disallowed when referee Willie Collum ruled goalkeeper Allan McGregor was fouled by Jon Daly in the build-up.
At the other end, Sone Aluko had a go from distance after collecting from David Healy but dragged the shot across the face of goal and wide of target.
The visitors still looked dangerous and Russell threatened again with a deflected effort that was claimed by McGregor, before the Rangers number one saved well from Paul Dixon at full-stretch.
But United were celebrating with 15 minutes gone when Gavin Gunning met a Gary Mackay-Steven cross and sent a firm header past McGregor from eight yards.
Rangers could have hauled themselves level almost immediately when Aluko skipped past Garry Kenneth and unleashed a shot that was palmed to safety by Dusan Pernis.
The United goalkeeper was then called into action to tip a strike from Healy around the post and deny the Northern Irishman the leveller, with the effort heading for the bottom corner.
The visitors scored again after 35 minutes when Dixon slipped the ball into the path of the unmarked Russell and he took advantage of the space to rifle high past the grasp of McGregor.
After a less than inspiring performance, Celik's involvement came to an end after just 45 minutes when he was replaced by Kerkar for the second half.
Little - whose injury problems meant his last game for Rangers was in October 2010 - then replaced Healy to partner Kerkar up front as the Ibrox side battled desperately to haul themselves back into the tie.
But the game could have been over as a contest had Kenneth been able to hit the target with a close-range header that was saved by the Rangers goalkeeper.
Little could have sparked a late comeback when he met an Aluko pass only to blast over the crossbar from six yards.
United then almost added to their lead when McGregor reacted quickly to block from John Rankin.