Cyriel Dessers got the Light Blues level with a deserved equaliser late in the game, but they were unable to hold on as Bruno Fernandes netted two minutes into injury time to secure the points for the home side.
There was a return for Jack Butland and Leon Balogun to the squad as they came back from injury to be named in the starting XI. Butland was between the sticks behind a back four of James Tavernier, Leon Balogun, Robin Propper and Jefte.
Connor Barron, Nico Raskin and Nedim Bajrami were in midfield, while Hamza Igamane led the line with support from Vaclav Cerny and Ridvan.
Manchester United headed into the match unbeaten in the league phase and looked to get off to a strong start at home with an early chance through Garnacho but it was held by Butland.
It was a good start for Rangers and Clement’s men had a fantastic opportunity inside the opening 10 minutes. Hamza Igamane picked the perfect pass to Nico Raskin who had made a run into the box, but the midfielder was unable to convert as Bayindir reacted quickly.
Shortly after James Tavernier whipped a fantastic delivery into the box which Ridvan got on the end of, but his shot was at the keeper and he gathered.
Garnacho came close for United as a clearance fell for the winger in space just inside the box but Nico Raskin got a header to divert it over the bar.
From the resulting corner, De Ligt headed home what Manchester United thought was an opening goal, but it was disallowed for a push on Robin Propper inside the box.
Jack Butland pulled off a fantastic save in the 36th minute as he blocked a shot from Joshua Zirkzee from inside the area, before Hamza Igamane’s long-range strike was deflected wide in what was an even 40 minutes at Old Trafford.
Butland was called into action again before the break as he got a strong hand to a left-footed strike from Lisandro Martinez to tip it over the bar.
There were two changes for Rangers at half-time as Bailey Rice and Ross McCausland were introduced in place of Leon Balogun and Connor Barron.
Rangers started the second-half in similar fashion to the first with a few opportunities in the opening five minutes. Bailey Rice had a curling shot go high of the target, before Vaclav Cerny forced a save from Altay Bayindir.
Manchester United looked threatening too and captain Bruno Fernandes almost found the breakthrough with a shot which was deflected wide by Robin Propper.
From the resulting corner, the home side had the opener as Jack Butland came out to punch away a Christian Eriksen corner but the goalkeeper mistimed the punch and the ball hit the Rangers number one and fell in the back of the net.
There was another change for Rangers with half an hour remaining as Findlay Curtis was introduced for his European debut in place of Vaclav Cerny.
Maguire headed wide from an Eriksen free-kick as the Premier League side looked to increase their advantage.
Cyriel Dessers was introduced for the final 20 minutes in Manchester as Rangers continued to search for an equaliser.
Butland again pulled off a good stop as he blocked a header from Manuel Ugarte.
With less than 10 minutes remaining, Rangers were continuing to push for an equaliser with Findlay Curtis picking out the run of Cyriel Dessers, but it was just wide of the left-hand post.
In the 88th minute Rangers had the deserved equaliser, and it was a fantastic finish from Cyriel Dessers. The striker beat the offside trap to run through on goal and take a superb touch before nesting the ball in the bottom corner.
However, two minutes into injury time, the home side were back ahead as captain Bruno Fernandes found the target and secured the victory for Manchester United.
RANGERS: Butland, Tavernier, Balogun (McCausland, 45), Propper, Jefte, Barron (Rice, 45), Raskin, Bajrami (Dessers, 70), Ridvan (King, 89), Cerny (Curtis, 59), Igamane.
Subs: Kelly, Munn, Lawrence, Dowell, Lovelace, Nsio, Hutton
Manchester United: Bayindir, De Ligt (Maguire, 45), Martinez, Fernandes, Zirkzee (Hojlund, 83), Eriksen (Mainoo, 74), Yoro (Malacia, 56), Amad, Garnacho, Dalot, Collyer (Ugarte, 74).
Subs: Heaton, Onana, Casemiro, Antony.