The Blades went ahead in the first half through a well-worked corner routine, which saw George Baldock dummy John Fleck’s centre and Ollie Norwood sidefoot home his first goal since January 2020.
The Blades held out for over 80 minutes to record a win that takes them back into the play-off places, and condemn Rangers to their sixth win in seven – further increasing the pressure on boss Warburton.
“It was a shocking goal to give away to a set piece we’ve practiced against,” he said.
“We didn’t start well and finished the half better. Second half was better but it was about having that bit of class in the final third and we didn’t show it.
“We had lots of ball. They’ve got a great record at home and we’ve lost to a shocking goal.”
Warburton admitted his side have lost some key players, with former Sheffield Wednesday goalkeeper Keiren Westwood signed as a free agent recently and playing 90 minutes against the Blades with Rangers’ other goalkeepers injured.
“But that’s football, you can’t bemoan that,” Warburton added,
“It gives opportunities to others to come in and show what they can do.
“I know Paul [Heckingbottom, United boss] has struggled here with Billy Sharp and Rhian Brewster and Oli McBurnie being out and everyone has struggled. We can’t afford to lose players for long periods of time.”