Rooney gave United a 1-0 lead at the break, diverting Javier Hernandez's shot past Ben Foster on the volley.
The United striker then made it 2-0 from the penalty spot, while West Brom's Jonas Olsson was earlier sent off for a second bookable offence.
Ferguson hails 'significant' win as Man Utd go top
It was a deserved win for United who should have added to their tally in the second half.
Victory took United top of the Premier League for the first time since early October and with 10 games remaining they will be favourites to claim their 20th league title following City's unexpected slip.
They head to Wolves next Sunday knowing a win would take them four points clear of City who do not play again in the Premier League until Wednesday week against Chelsea.
Manager Sir Alex Ferguson used his programme notes to rally both United players and supporters as his side entered the "business end" of the campaign.
And Ferguson picked a side full of attacking intent with Rooney, Danny Welbeck and Hernandez all named in the starting line-up.
Steve WilsonBBC Match of the Day commentator
It took United a while to get going in this game, but this was a victory that was thoroughly deserved. Wayne Rooney gambled and got his reward, it was a fine finish.
But their start to the game was turgid and lacklustre, with West Brom the better of the two sides early on.
Keith Andrews poked the ball wide at full stretch following Liam Ridgewell's cross, and later forced a fine low save from David de Gea.
But the champions soon got going, with Paul Scholes finding space to orchestrate things from midfield, and in Ashley Young, United had the game's outstanding player.
The pressure eventually told, with Rooney breaking the deadlock. United's in-form striker gambled at the near post and expertly diverted Hernandez's shot past Foster on the volley from six yards out.
It was Rooney's 25th goal of the season - and his 14th in as many games - and gave them a deserved lead at the break.
'Always an uphill task', admits Hodgson
West Brom began the second half like the first - full of intent - and were unlucky not to have been awarded a penalty after Patrice Evra tripped Peter Odemwingie in the box.
Referee Lee Probert waved away their appeals and replays suggested the Baggies man may have gone to ground too easily.
Straight after, United nearly doubled their advantage with Hernandez crashing a fierce shot from the edge of the box against the post before West Brom cleared for a corner.
From the resulting corner Rooney went close with his adventurous overhead kick straight at Foster.
United's pressure continued and Welbeck should have put them 2-0 up but, after rounding Foster, he inexplicably fired wide of an empty goal from a tight angle.
2 Apr/31 Mar
West Brom (h)
Aston Villa (h)
Man City (a)
Man Utd (h)
West Brom's task became that much harder with Olsson shown a second yellow for a lunge on Hernandez, meaning the Baggies had to play with 10 men for the remaining 25 minutes.
Young continued to terrorise the Baggies defence and he was bundled over by Andrews for a United penalty, which Rooney converted for 2-0.
With the game won, United introduced highly rated youngster Paul Pogba, 18, for Scholes and it provided an interesting sideshow for the remaining 20 minutes, as Pogba has been linked with a
summer move to Juventus.
The Baggies brought on Shane Long and he forced an excellent save from De Gea while, at the other end, Young headed wide from 12 yards.
But it stayed at 2-0 with the biggest cheer of the day reserved for the announcement of Swansea's shock victory over Manchester City.
West Brom boss Roy Hodgson said:
"We made a good fist of the first half until Rooney's goal.
"I didn't think we were attacking as well as we could do in that period. When we won the ball back we lost it too quickly and easily.''
Full Time The referee blows for the end of the match.
Javier Hernandez has an effort at goal from deep inside the area missing to the left of the target.
Shane Long concedes a free kick for a foul on Phil Jones. Ashley Young takes the free kick.
Michael Carrick takes a shot. Blocked by Gareth McAuley.
Inswinging corner taken by Nicky Shorey, save by David De Gea.
Phil Jones produces a cross, Gareth McAuley manages to make a clearance.
Ashley Young decides to take the corner short.
Shot from just outside the area by Paul Pogba misses to the right of the target.
The ball is sent over by Phil Jones, Javier Hernandez takes a shot from inside the six-yard box clearing the crossbar.
The ball is crossed by Phil Jones.
The ball is delivered by Javier Hernandez, Header by Ashley Young from deep inside the penalty area misses to the left of the target.
Shane Long takes a shot. Save by David De Gea. Inswinging corner taken by Nicky Shorey from the right by-line, clearance by Paul Pogba.
Shane Long challenges Jonathan Evans unfairly and gives away a free kick. David De Gea takes the free kick.
Nicky Shorey takes a inswinging corner to the near post, clearance made by Tom Cleverley.
Substitution Tom Cleverley on for Danny Welbeck.
Substitution Shane Long comes on in place of Marc-Antoine Fortune.
Javier Hernandez has an effort from just inside the box that misses to the left of the goal.
Centre by Phil Jones, Nicky Shorey gets a block in.
Substitution Paul Pogba on for Paul Scholes.
Substitution Paul Scharner is brought on as a substitute for Keith Andrews.
Substitution Nicky Shorey comes on in place of Chris Brunt.
Ashley Young provided the assist for the goal.
70:32 Goal scored
Goal! - Wayne Rooney - Man Utd 2 - 0 West Brom Wayne Rooney scores a placed penalty. Man Utd 2-0 West Brom.
Ashley Young fouled by Keith Andrews, the ref awards a Penalty.
Patrice Evra takes a shot. Blocked by Gareth McAuley. Ashley Young takes a shot. Ben Foster makes a save. The referee blows for offside against Javier Hernandez. Ben Foster takes the indirect free kick.
Michael Carrick takes the direct free kick.
Dismissal The referee red cards Jonas Olsson for second bookable offence.
Unfair challenge on Javier Hernandez by Jonas Olsson results in a free kick.
Paul Scholes takes a shot. Ben Foster makes a save.
Ashley Young produces a cross, Ben Foster makes a save.
Patrice Evra delivers the ball, Gareth McAuley gets a block in. Ashley Young takes a inswinging corner from the left by-line to the near post, clearance by Jonas Olsson.
Shot from 35 yards from Gabriel Tamas. David De Gea makes a save.
Shot from inside the six-yard box by Danny Welbeck goes wide right of the goal.
Michael Carrick gives away a free kick for an unfair challenge on James Morrison. Free kick taken by Gabriel Tamas.
Phil Jones delivers the ball, save by Ben Foster.
The ball is delivered by Phil Jones, save by Ben Foster.
The official flags Marc-Antoine Fortune offside. David De Gea restarts play with the free kick.
Javier Hernandez produces a right-footed shot from just inside the area and hits the post. Inswinging corner taken from the left by-line by Ashley Young. Shot by Wayne Rooney. Save by Ben Foster.
The assistant referee signals for offside against Patrice Evra. Ben Foster restarts play with the free kick.
Rio Ferdinand restarts play with the free kick.
Booking The referee cautions Jonas Olsson for unsporting behaviour.
Wayne Rooney fouled by Jonas Olsson, the ref awards a free kick.
Foul by Jonas Olsson on Jonathan Evans, free kick awarded. Michael Carrick restarts play with the free kick.
Chris Brunt sends in a cross, Phil Jones gets a block in. Corner taken by James Morrison from the left by-line played to the near post, Danny Welbeck manages to make a clearance.
The match restarts for the second half.
45:00 +2:02 Half time
Half Time The referee calls an end to the first half.
Free kick awarded for a foul by Phil Jones on Peter Odemwingie. Free kick crossed left-footed by Chris Brunt.
Ashley Young sends in a cross, clearance made by Keith Andrews. Corner from the right by-line taken by Ashley Young, clearance made by Marc-Antoine Fortune.
Foul by Chris Brunt on Phil Jones, free kick awarded. Paul Scholes takes the free kick.
Marc-Antoine Fortune is adjudged to have handled the ball. Paul Scholes takes the direct free kick.
Wayne Rooney takes a shot. Gareth McAuley gets a block in. The offside flag is raised against Patrice Evra. Free kick taken by Ben Foster.
A cross is delivered by Phil Jones, Liam Ridgewell makes a clearance.
The ball is swung over by Danny Welbeck, clearance by Gabriel Tamas.
Chris Brunt challenges Danny Welbeck unfairly and gives away a free kick. Phil Jones takes the free kick.
Ashley Young sends in a cross, Header from close range by Phil Jones misses to the right of the goal.
Foul by Gabriel Tamas on Ashley Young, free kick awarded. Ashley Young restarts play with the free kick.
Assist by Javier Hernandez.
34:58 Goal scored
Goal! - Wayne Rooney - Man Utd 1 - 0 West Brom Wayne Rooney scores a goal from inside the area to the bottom left corner of the goal. Man Utd 1-0 West Brom.
Javier Hernandez takes a shot. Wayne Rooney gets a block in.
Corner taken by Ashley Young, Jonas Olsson makes a clearance.
The ball is crossed by Ashley Young, Wayne Rooney takes the chance to get an effort at goal. Comfortable save by Ben Foster.
Javier Hernandez is caught offside. Ben Foster takes the free kick.
The ball is delivered by Marc-Antoine Fortune, clearance by Jonathan Evans.
Ashley Young takes a short corner, The ball is swung over by Michael Carrick, Gabriel Tamas gets a block in. Inswinging corner taken by Ashley Young, Header from close range by Danny Welbeck clears the bar.
Patrice Evra sends in a cross, Gareth McAuley makes a clearance.
Foul by Marc-Antoine Fortune on Phil Jones, free kick awarded. Direct free kick taken by Jonathan Evans.
Javier Hernandez is ruled offside. Ben Foster takes the direct free kick.
Jonathan Evans concedes a free kick for a foul on Marc-Antoine Fortune. Free kick taken by Chris Brunt. Effort on goal by Keith Andrews from 25 yards. Save made by David De Gea. Corner taken by James Morrison from the left by-line to the near post, Header from close range by Liam Ridgewell misses to the right of the goal.
Peter Odemwingie challenges Patrice Evra unfairly and gives away a free kick. David De Gea takes the free kick.
Unfair challenge on Liam Ridgewell by Javier Hernandez results in a free kick. Liam Ridgewell takes the direct free kick.
Wayne Rooney produces a right-footed shot from just outside the penalty area that goes harmlessly over the target.
Keith Andrews concedes a free kick for a foul on Paul Scholes. Paul Scholes takes the direct free kick.
Ashley Young takes a shot. Save by Ben Foster.
Free kick awarded for an unfair challenge on Youssouf Mulumbu by Paul Scholes. Direct free kick taken by Chris Brunt. The ball is delivered by Jonas Olsson, clearance by Jonathan Evans. James Morrison takes a shot from just inside the penalty box which clears the crossbar.
Free kick awarded for an unfair challenge on Patrice Evra by James Morrison. Ashley Young crosses the ball in from the free kick, Chris Brunt makes a clearance.
Liam Ridgewell delivers the ball, Close range effort by Keith Andrews misses to the left of the goal.
Corner taken by Ashley Young played to the near post, Keith Andrews makes a clearance.
Ashley Young delivers the ball, blocked by Youssouf Mulumbu.
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