DaMarcus Beasley put Rangers in front as the Gers enjoyed a comfortable victory, much to the frustration of United manager Craig Levein.
Levein said: "I didn't feel we had enough players who played anywhere near the level they have been playing at, to enable us to win the match.
"To win tonight's match we needed to be playing well and needed everybody on top form and that didn't happen.
"That's an example of what can happen when we're not playing at our best.
"There was a big crowd and support right behind us and I can only apologise to them.
"We struggled from the first minute to the last and all credit to Rangers, they certainly played well enough to beat us comfortably tonight."
Levein admitted he had taken his team to task at the interval, after their lacklustre start to the match.
"I had a go at half-time and it didn't make an awful lot of difference in the second half," he said.
Rangers boss Walter Smith said of the result: "It was obviously very pleasing. I was delighted to get the win.
"It's a difficult place to come and get a win.
"It's been difficult for us over the last few seasons and tonight I felt we played extremely well and certainly deserved to win."