Rafa Benitez started with the same 11 that hammered Marseille in midweek as he looked to engineer his illusive first league victory over United.
And his side started brightly, carving out the first shot on goal in the opening minutes but Sami Hyypia's tame effort looped over the bar.
Fernando Torres then tried his luck after a defensive error from Rio Ferdinand but his effort curled wide of Edwin Van der Sar's post.
Apart from that, chances were scarce in the opening 25 minutes and the football wasn't pretty as the teams flew at each other at a ferocious pace.
But Liverpool should have been ahead in the 26th minute when Harry Kewell's shot was cleared off the line by Anderson and Torres headed the rebound wide with the goal gaping.
It was a bitterly cold day in Anfield but the atmosphere was red hot as the tackles flew in.
And it was Liverpool who looked like opening the scoring when John Arne Riise came close by shooting wide after the ball fell to his powerful left foot.
Then on 36 minutes, Torres volleyed wide from Steven Gerrard's deflected free-kick as the home side threatened again.
Sir Alex Ferguson's men had shown little but ended the half strongly with Cristiano Ronaldo and Ryan Giggs stretching Liverpool down the flanks.
And with seconds remaining before half-time they landed the killer blow through Tevez.
Wayne Rooney was left unmarked on the edge of the box following a corner and the striker's low drive fell to the Argentinian who flicked the ball into the roof of the net to give his side an unlikely lead.
The goal stunned Liverpool and they started the second half sloppily, giving the ball away needlessly as United pressed.
But Benitez's men started to come back into it as Ryan Babel replaced the ineffective Kewell and Peter Crouch came on for the out of sorts Dirk Kuyt.
First Torres found Gerrard on the edge of the box but the skipper screwed his shot wide.
Then Dutchman Babel hit a fierce low shot that fizzed just wide of the post as Liverpool searched desperately for an equaliser.
But United, dangerous with another counter attack, should have put their opponents to bed with ten minutes to go.
Ronaldo skipped past Hyypia before finding Rooney, but the striker somehow shot wide with just Jose Reina to beat.
Liverpool continued to push forward but Ferguson's defence was outstanding with Ferdinand, Patrice Evra and Nemanja Vidic throwing themselves at everything.
Gerrard hit the top of the net late on with a free-kick, but United held firm for another famous victory at Anfield.