United, the only unbeaten side in English football's top flight this term and who've two games in hand, remained top on goal difference alone after Manchester City's 1-0 victory at home to Blackpool on Saturday.
"It's a massive result for us," United manager Sir Alex Ferguson told Sky Sports. "Over the years we've had to battle out games and we had to do that today."
At Anfield, Joe Cole's last-minute winner gave under pressure Liverpool manager Roy Hodgson a desperately needed come from behind 2-1 victory over Bolton Wanderers.
"I don't know about relief, I just thought we played so well," said Hodgson. "It showed the character of the players and their determination to turn things around for the club and give us a better 2011 than 2010.
"Anything other than a victory would have been an unfair result."
West Ham moved off the bottom of the table and out of the relegation zone with a 2-0 win at home to drop rivals Wolves, who were left propping up the Premier League just days after their shock win away to Liverpool.
If the top two spots are an all Manchester affair, third and fourth belonged to north London after Arsenal and Tottenham Hotspur both won on Saturday.
Arsenal stayed third, two points off the summit, after Robin van Persie's 13th minute deflected free-kick -- the Dutch striker's first league goal this season -- Samir Nasri's second-half goal and Roger Johnson's 66th minute own-goal saw the Gunners win 3-0 away to Birmingham in Saturday's late kick-off.
Victory was just what Arsenal needed after a midweek draw with Wigan took some of the shine off their 3-1 humbling of champions Chelsea.
"I am delighted because we have shown physical qualities, we were up for the challenge and mentally as well we kept our calm," Arsene Wenger, the Arsenal manager, told ESPN.
"We kept faith in our game, that we could put the ball down and play our game. I feel we have had a great performance."
Spurs leapfrogged Chelsea, who can regain fourth spot against Aston Villa on Sunday, after Gareth Bale's header shortly before half-time meant they won 1-0 at home to relegation-threatened Fulham.
Rooney opened the scoring at The Hawthorns but substitute Javier Hernandez, who netted both goals in United's other away league win this term, a 2-1 success at Stoke in October, sealed victory 15 minutes from time by heading in the forward's corner after the Baggies' Peter Odemwingie had missed a penalty.
England striker Rooney also played through an ankle injury late on when Ferguson had used up all three of his substitutes.
"Wayne Rooney was absolutely out of this world," Ferguson said. "He's off the mark and, hopefully he's on a (goal) run now."
Rooney put United 1-0 up as early as the third minute at The Hawthorns when he stooped to head in a Patrice Evra cross.
But West Brom were level 11 minutes later when James Morrison lashed the ball in from 25 yards out.
At Eastlands, Adam Johnson's 34th-minute goal proved decisive in a match where City striker Carlos Tevez missed a subsequent penalty.
Hodgson, who apologised to Liverpool fans on Friday after criticising their lack of support following the Wolves match, saw his side fall behind when Bolton's Kevin Davies headed in Matt Taylor's free-kick shortly before half-time.
However, Liverpool were level early in the second half when Steven Gerrard fed Fernando Torres and the Spain striker volleyed home.
They then snatched victory when Cole turned in a Maxi Rodriguez centre.
West Ham took a fortunate lead when Frederic Piquionne's cross went in off the knee off Wolves defender Ronald Zubar in the 51st minute at Upton Park before Freddie Sears sealed victory 11 minutes from time.
Stoke left Everton looking over their shoulders following a 2-0 win secured by Kenwyne Jones's first half header and Phil Jagielka's 69th minute own-goal.
Sunderland won 3-0 at home to Blackburn Rovers after first half goals from Danny Welbeck and England striker Darren Bent were topped off by Asamoah Gyan's 89th minute effort.
Wigan face Newcastle on Sunday.
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