United has now won eight games in a row -- six by 1-0 scores.
"This has been a difficult match for us over the years and this was a great result for us to come away with a win," said Dutchman goalkeeper Edwin van der Sar, whose shutout streak is 1,212 minutes, a British league record. "The defenders are incredible, how they defend, how they clear the ball, it's a team effort."
Liverpool twice came back from one goal down on goals by Fabio Aurelio and Dirk Kuyt before Spaniard Fernando Torres won the game in stoppage time.
Surging Aston Villa moved up to third place, three points behind Liverpool, with a 2-0 win at Blackburn.
Chelsea and Arsenal both had to settle for 0-0 ties.
Chelsea, now fourth seven points off the pace set by Manchester United, tied Hull City, 0-0, at Stamford Bridge, while Arsenal is a distant fifth, five points behind the Blues, after drawing, 0-0, at north London rival Tottenham.