Lee Wilkie opened the scoring for United in the first half, before Daniel Cousin restored parity from the penalty spot shortly after the break.
But Rangers' joy lasted just two minutes before the hosts reclaimed their lead also through a penalty from Barry Robson.
"I am obviously pleased," Levein said. "The scoreline was the same as it was a year ago but today's result doesn't tell the real story - Rangers are a vastly improved team now and we needed to be too.
"The players did everything I asked of them and, in some cases, a little bit more."
The win allowed United to move level on points with Rangers - just three behind leaders Celtic - but Levein is refusing to look too far ahead in terms of what his players could achieve this season after suffering their fair share of disappointment in the past.
He said: "We have still got a long way to go - 27 games - so it is important to concentrate on the little things in training and keep improving on the pitch. Hopefully if we do that, it will keep us where we are."
Gers boss Walter Smith said: "The European game had nothing to do with it.
"The players had plenty of time to recover from the game on Tuesday night. It certainly wasn't tiredness and we saw that in the second half."