Tom Lawrence’s seventh minute strike was the difference maker for Clement’s side, who boasted the opportunities to kill the game but were forced to dig deep in the second half to keep the home side at arm’s length.
It was a third clean sheet of the campaign for the Light Blues, who can now begin to look forward to next weekend’s return to Ibrox in the Premier Sports Cup against the blue side of Dundee.
The Gers were without Vaclav Cerny for the trip to Tayside due to illness, with boss Clement handing starts to Kieran Dowell and the returning Oscar Cortes.
There were spots on the bench, too, for deadline day arrivals, Nedim Bajrami and Neraysho Kasanwirjo, as Tom Lawrence was restored to the side from the Light Blues’ previous league outing.
And Clement’s selection was justified inside five minutes when Lawrence produced an exquisite piece of individual play after combining with Cortes and Cyriel Dessers in midfield, slaloming his way into the Dundee United penalty area before dinking a delicious finish over the onrushing Jack Walton.
Rangers were rampant in the opening exchanges and the opener was a further shot in the arm.
Jefte was next to trouble the home side’s penalty area when an early delivery from the Brazilian flashed across the face of goal without any takers in white and blue.
A quarter of an hour had elapsed when Dowell, who was the subject of praise from the manager before the game, stung the palms of Walton from 25 yards out and the United goalkeeper gathered at the second attempt with Dessers lurking with intent.
Rangers continued to move through the gears in the first half and Lawrence was perhaps guilty of delaying his pass a split-second too long with Cortes making inroads on the left-hand side, with the Colombian curling an effort into the hands of Walton from a tight angle.
The Light Blues came close to adding a second of the afternoon on 23 minutes when Connor Barron popped up inside the penalty and fed a pass inside to Lawrence whose first-time effort unfortunately cannoned back off Dessers who was positioned, as all good strikers enjoy, in the middle of the goal.
The away side’s dominance in the contest was reflected midway through the first half when Jim Goodwin withdrew the experienced Kevin Holt in place of Ross Docherty, switching to a 4-3-3 in an attempt to gain a foothold in the encounter.
Although the hosts were doing very little in attacking sense to trouble Jack Butland, the change in system stemmed the flow of the Gers attack and an effort from Dessers on the stroke of the interval, saved by Walton at his near post, was all that Rangers had to show for the remainder of the opening period.
The half-time period appeared to reinvigorate Rangers, who emerged from the interval and had Walton in the United goal worried when the ball dropped to Lawrence on the edge of the penalty area.
Unfortunately for the visitors, the Welsh international shot straight down the throat of the United goalkeeper, but it was certainly a mark of intent from Clement’s side.
However, United would fashion their most promising opportunity of the contest on 55 minutes when Kristijan Trapanovski cut inside off the left and his effort whisked the outside of the far post.
The momentum, it seemed, was with the hosts for the first time in the contest, who were deriving particular joy through the dangerous Trapanovski on the left-hand side.
Lawrence, however, was the bright spark on the pitch and he almost doubled his tally when he weaved his way through several United bodies and bent an effort only inches wide of Walton’s goal.
A second goal was worth its weight in gold for Clement’s side and they were served a warning sign when Ross Graham rose highest from a corner and glanced a header just wide of Butland’s goal.
Opportunity then beckoned for Rangers moments later when substitute Hamza Igamane lifted his head and fed a pass through to Lawrence beyond the United backline, but the midfielder was denied by the outstretched legs of Walton.
Igamane was inches away from his first Gers goal when he met a corner at the near post which flew into the side netting.
Despite the home side’s best efforts, the Light Blues were able to see the contest with a minimum of fuss to preserve a priceless three points in their league campaign.
DUNDEE UNITED: Walton, Holt (Docherty, 32), Graham, Trapanovski, Babunski (Dalby, 75), Ferry, Sibbald, Adegboyega (Sevelj, 45), Stephenson, van der Sande (Moult, 75), Gallagher
Subs: Richards, Odada, Middleton, Thomson, Ubochioma
RANGERS: Butland, Tavernier, Souttar, Propper, Jefte, Barron, Diomande (Raskin, 72), Dowell (Sterling, 72), Lawrence, Cortes (Bajrami, 45), Dessers (Igamane, 62)
Subs: Kelly, Kasanwirjo, Balogun, McCausland, Fraser