Robin van Persie had given United a first-half lead and Nemanja Vidic extended their dominance in fortuitous fashion early in the second half.
But substitute Daniel Sturridge’s first league goal for Liverpool brought the visitors back into the game and there were one or two nervy moments before the Red Devils confirmed the three points.
United took the lead with the opening chance of the game on 19 minutes when van Persie swept home first time from Patrice Evra’s low cross following a patient build-up by the leaders.
Van Persie was almost in again on 27 minutes after Ashley Young’s pass found him in space just inside the box, but the Dutchman’s first-time finish on this occasion flew over the bar.
Joe Allen was enduring a difficult afternoon and the midfielder’s loose pass was seized upon by Danny Welbeck before Daniel Agger produced a great block to deny the striker.
Luis Suarez blazed over Liverpool’s only shot of the first half before Tom Cleverley flashed a volley wide after Allen’s poor headed clearance from Young’s cross.
The Red Devils should have extended their lead just before the break, but van Persie’s flick from Rafael’s cross was cleared off the line by Martin Skrtel and Pepe Reina was hurt in a collision with Shinji Kagawa in the scramble for the loose ball.
United struck again on 53 minutes when Skrtel was booked for hauling back Welbeck and van Persie’s delivery was met by Evra, whose header was inadvertently deflected in by Vidic.
Liverpool were quick to respond and pulled a goal back within four minutes when Sturridge, a half-time replacement for Lucas, was sharpest to the loose ball after David de Gea could only parry Steven Gerrard’s low drive from outside the box.
Kagawa forced a flying save out of Reina with a deft curler before Glen Johnson, Sturridge and another substitute Fabio Borini all failed to find the target in a five-minute spell of Liverpool pressure.
Sturridge had Liverpool’s best chance in the closing stages, but scooped his shot over the bar and United spent the last couple of minutes of stoppage-time deep in opposition territory before Howard Webb blew for full-time.