The depleted visitors, who had four teenagers in their line-up, held firm for more than 70 minutes before captain James Tavernier slammed in a penalty.
The result stretches Rangers’ advantage to seven points, but League Cup finalists Celtic have a game in hand.
United, who have one win in their last nine, drop to fifth.
However, it was a hugely impressive effort from Thomas Courts’ men, who had an entire XI of first-team regulars – including the likes of Benjamin Siegrist, Jeando Fuchs, Ian Harkes, Peter Pawlett, Ilmari Niskanen and Marc McNulty – missing for the trip.
But Giovanni van Bronckhorst’s side showed the mark of champions and found a way to win, ensuring the Dutchman’s winning league start continues.
Travelling to Ibrox will a full squad would have still been a daunting assignment for United, given the run of form they have endured, coupled with the fact Rangers are unbeaten at home in the league since March 2020.
But with four teenagers in a depleted starting XI, getting any kind of positive result was bordering on insurmountable for Courts when Alan Muir blew the first whistle.
It wasn’t just in the United line-up there was eye-catching team news, though, as Rangers right-backs Tavernier and Nathan Patterson started together for the first time.
Captain Tavernier began at centre-back, with Patterson in his natural position, but the experiment only lasted a matter of minutes as Tavernier was beaten far too easily by Adrian Sporle in the build-up to an effort on goal from Nicky Clark.
Courts’ youngsters were shining in the Ibrox fog, as 18-year-old Chris Mochrie had a goalbound close-range shot blocked.
Rangers showed fleeting signs of clicking into gear in the first half, with Alfredo Morelos dragging a finish wide and forcing Trevor Carson to tip a header over either side of efforts from Glen Kamara, John Lundstram and Tavernier.
But too often their play was safe and slack in a largely pedestrian first-half showing. Scott Wright’s half-time introduction for Lundstram gave the hosts more in attack as Rangers turned the screw.
Morelos drilled over on the turn before neat play in the box between the Colombian and Wright resulted in the former Aberdeen man forcing Carson into a fine stop.
Then came the penalty, as Scott McMann’s clear handball allowed captain Tavernier to step up and dispatch to Carson’s right.
Rangers passed up chances to settle to game and left the door open for a spirited United, who gave their hosts a fright when teenager Darren Watson headed off the bar in injury time after poor defending from the hosts.
But Van Bronckhorst’s men survived the scare to keep momentum rolling while turning up the heat on Celtic.
Man of the match – Scott Wright
What did we learn?
When the teams were announced, you would be forgiven for expecting a demoralising day for United. A fourth consecutive defeat will hurt, of course, but Courts can take huge pride in his young side for going toe to toe with the league leaders.
The United boss has giving youth a chance high on his agenda, and performances like that will give him encouragement to throw them in again.
Van Bronckhorst wasn’t mucking about when he hooked Lundstram at the break. Wright’s introduction was the catalyst for an improved second-half display and the Dutchman must take credit for that.
The Ibrox side’s last league defeat, a 1-0 loss at Tannadice in August, feels a long time ago now and has been followed by 17 Premiership games. When leagues are won, games like these are considered the ones that pushed teams over the line.
What did they say?
Dundee United head coach Thomas Courts: “The performance and result can sometimes spiral at venues like this, but they [young players] really represented the club fantastically well in what’s been a really tough week.
“I’m really proud of them. Barring a penalty incident, I don’t think there was too much in the game. They really have to take a big stride forward and experience this on a regular basis.”
Both teams resume Premiership action on 26 December as Rangers host St Mirren and Dundee United welcome Hibernian (both 15:00 GMT).