Manchester United bounced back from last Sunday's 6-1 defeat by Manchester City as Javier Hernandez's goal gave them a hard-earned victory at Everton.
Hernandez was well-placed just two yards out to side-foot in Patrice Evra's cross in the 19th minute.
The goal was reward for a bright start from United, but they were matched by Everton for the remainder of the game.
However, Leighton Baines struck the crossbar with a free-kick and Louis Saha's shooting lacked precision.
A point would have been just reward for the efforts of David Moyes' side, but having failed to earn that they can take a lot of heart from a spirited display against opponents with a point to prove following a week of largely negative headlines.
Moyes praises players despite defeat
For United, victory was all-important, with a stellar performance second, following their home humbling last weekend. They achieved the former, with only brief but impressive flashes of the latter.
There had been much debate this week over how United manager Sir Alex Ferguson would react to last Sunday's defeat in terms of team selection.
In the event, he dropped defender Rio Ferdinand and winger Nani, retained Jonny Evans, despite his red card against City, and restored Nemanja Vidic and Tom Cleverley to what was an attacking line-up, featuring Hernandez, Danny Welbeck and Wayne Rooney. The latter's every touch was greeted by the now standard jeers from the Everton support.
The approach initially paid dividends as United made positive and fluent early forays into Everton territory, one of which teed up Park Ji-sung, whose six-yard shot was saved by Tim Howard.
United capped their impressive start by taking the lead after 19 minutes, when Evra's pin-point cross from the left meant Hernandez had only to side-foot home from two yards out.
However, instead of kicking on, United shrank somewhat and Everton grew as a force in the game.
Everton have won only one of their last 13 meetings with Manchester United
This was Manchester United's first Premier League win on Merseyside in almost four years
David de Gea was forced to save a Leon Osman snapshot and Jack Rodwell fired wide from 25 yards before Baines struck the crossbar with a free-kick from the edge of the area as the home side pressed hard for an equaliser.
Hernandez could have made the second half easier for his side had he finished a well-constructed move with his shot from the edge of the box, but Howard's save set up a tense second 45 minutes.
United's growing nervousness in defence was clear to see as Saha twice tested De Gea with shots from the edge of the area.
The visitors then almost gifted Everton an equaliser when Evans gave away possession. Fortunately for the defender, Rodwell's shot from an angle flew just past the far post.
With half an hour to go, Welbeck could have settled United nerves but his shot from close range, on an angle, was palmed over by Howard.
This left United needing to show resolution and composure in the final minutes of the match in the face of further pressure from the home side and - barring a home penalty appeal for a coming-together between substitute Magaye Gueye and Evra - they did so with aplomb.
Everton manager David Moyes:
"I think we are unlucky not to get at least a point out of the game. We played well and Manchester United had few attempts, especially in the second half. I thought we looked the stronger side in the second half.
"Maybe it was just that bit of quality in the final third that we didn't have.
"Great credit to the lads. We have gone toe to toe with Chelsea [in the Carling Cup] in midweek, when we were unlucky not to go through, and done the same today."
Manchester United manager Sir Alex Ferguson:
Ferguson delighted to get important win
"It was a hard game and we expected that because every time you come to Everton the crowd get behind them and show great enthusiasm.
"But in the main, in the first half particularly, we did very well and in the second half we defended well and that was the pleasing aspect for me.
"The fact that there have been so many chances against us recently - we had to reduce that if we are going to win the league and today we didn't have any problems."
Apostolos Vellios takes a shot. Save by David De Gea.
Free kick awarded for a foul by Jonathan Evans on Apostolos Vellios. Direct free kick taken by Phil Jagielka.
Darren Fletcher restarts play with the free kick.
90:00 +4:23 Booking
Booking Marouane Fellaini receives a caution for unsporting behaviour.
Marouane Fellaini concedes a free kick for a foul on Phil Jones.
Leighton Baines concedes a free kick for a foul on Antonio Valencia. Phil Jones takes the direct free kick.
Dimitar Berbatov gives away a free kick for an unfair challenge on Johnny Heitinga. Johnny Heitinga takes the free kick.
Seamus Coleman delivers the ball.
Inswinging corner taken left-footed by Magaye Gueye from the right by-line to the near post, clearance by Luis Nani.
The ball is crossed by Luis Nani, clearance by Apostolos Vellios.
Marouane Fellaini challenges Luis Nani unfairly and gives away a free kick. Wayne Rooney takes the direct free kick.
Marouane Fellaini is caught offside. Indirect free kick taken by David De Gea.
Seamus Coleman takes a shot. Blocked by Patrice Evra.
Substitution Magaye Gueye joins the action as a substitute, replacing Louis Saha.
A cross is delivered by Antonio Valencia, Leighton Baines gets a block in.
Seamus Coleman takes a shot. Comfortable save by David De Gea.
Foul by Antonio Valencia on Ross Barkley, free kick awarded. Direct free kick taken by Johnny Heitinga.
Inswinging corner taken by Leighton Baines from the right by-line played to the near post. Louis Saha takes a shot. Nemanja Vidic gets a block in. Jack Rodwell has an effort at goal from just outside the box which goes wide of the right-hand upright.
Tony Hibbert produces a cross, Nemanja Vidic manages to make a clearance.
Louis Saha produces a left-footed shot from outside the penalty box and misses right.
Substitution Leon Osman leaves the field to be replaced by Apostolos Vellios.
Shot from just outside the box by Phil Jagielka goes over the target.
Johnny Heitinga takes the free kick.
Substitution Antonio Valencia comes on in place of Danny Welbeck.
Dimitar Berbatov concedes a free kick for a foul on Phil Jagielka.
Free kick awarded for a foul by Johnny Heitinga on Ji-Sung Park. Darren Fletcher takes the direct free kick.
Luis Nani crosses the ball, save by Tim Howard.
Free kick awarded for an unfair challenge on Ji-Sung Park by Marouane Fellaini. Free kick taken by Phil Jones.
Seamus Coleman produces a left-footed shot from just outside the box that goes wide left of the goal.
Dimitar Berbatov is caught offside. Free kick taken by Tim Howard.
Dimitar Berbatov challenges Leighton Baines unfairly and gives away a free kick. Johnny Heitinga restarts play with the free kick.
Booking Seamus Coleman receives a yellow card for unsporting behaviour.
Substitution Dimitar Berbatov is brought on as a substitute for Javier Hernandez.
Luis Nani delivers the ball, Tim Howard makes a save.
Luis Nani takes a short corner.
Danny Welbeck takes a shot. Tim Howard makes a save.
Free kick awarded for a foul by Tony Hibbert on Javier Hernandez. Jonathan Evans restarts play with the free kick.
Substitution Luis Nani on for Tom Cleverley.
Phil Jagielka sends in a cross, Nemanja Vidic manages to make a clearance.
Centre by Seamus Coleman, Nemanja Vidic manages to make a clearance.
Jack Rodwell takes a shot. Save by David De Gea.
Louis Saha is ruled offside. David De Gea takes the free kick.
The referee penalises Javier Hernandez for handball. Phil Jagielka takes the free kick.
Marouane Fellaini gives away a free kick for an unfair challenge on Darren Fletcher. Direct free kick taken by Jonathan Evans.
Tony Hibbert crosses the ball.
Louis Saha challenges Phil Jones unfairly and gives away a free kick. Direct free kick taken by Tom Cleverley.
Free kick awarded for a foul by Javier Hernandez on Phil Jagielka. Free kick taken by Phil Jagielka.
The referee blows his whistle to start the second half.
Substitution Ross Barkley is brought on as a substitute for Diniyar Bilyaletdinov.
45:00 +1:15 Half time
Half Time The match has reached half-time.
Louis Saha takes a shot. Save by David De Gea.
The ball is crossed by Diniyar Bilyaletdinov.
Louis Saha takes a shot. Comfortable save by David De Gea.
Javier Hernandez takes a shot. Comfortable save by Tim Howard.
The referee penalises Seamus Coleman for handball. Free kick crossed right-footed by Tom Cleverley from left wing, clearance made by Tony Hibbert.
Unfair challenge on Marouane Fellaini by Darren Fletcher results in a free kick. Shot on goal comes in from Leighton Baines from the free kick,
The referee blows for offside. Free kick taken by David De Gea.
Shot from just outside the penalty box by Jack Rodwell misses to the left of the target.
Leon Osman takes a shot. David De Gea makes a save.
The ball is delivered by Diniyar Bilyaletdinov, clearance by Jonathan Evans.
Nemanja Vidic concedes a free kick for a foul on Leon Osman. Leighton Baines restarts play with the free kick. Louis Saha takes a shot. Javier Hernandez gets a block in.
Unfair challenge on Danny Welbeck by Tony Hibbert results in a free kick. Free kick taken by Tom Cleverley.
Leighton Baines takes the inswinging corner, clearance made by Danny Welbeck.
Diniyar Bilyaletdinov takes a shot. Blocked by Darren Fletcher.
Darren Fletcher takes the direct free kick.
Booking Jack Rodwell goes into the book for unsporting behaviour.
Phil Jones fouled by Jack Rodwell, the ref awards a free kick.
Patrice Evra takes a shot. Tony Hibbert gets a block in.
Danny Welbeck takes a shot. Phil Jagielka gets a block in.
Assist on the goal came from Patrice Evra.
18:16 Goal scored
Goal - Javier Hernandez - Everton 0 - 1 Man Utd Javier Hernandez fires in a goal from inside the six-yard box to the bottom right corner of the goal. Everton 0-1 Man Utd.
Wayne Rooney has shot on goal from just outside the box which goes wide of the right-hand upright.
Javier Hernandez is flagged offside by the assistant referee. Johnny Heitinga takes the free kick.
The official flags Javier Hernandez offside. Tim Howard takes the indirect free kick.
Seamus Coleman gives away a free kick for an unfair challenge on Patrice Evra. Darren Fletcher takes the direct free kick.
Free kick awarded for a foul by Marouane Fellaini on Ji-Sung Park. Darren Fletcher takes the direct free kick.
The assistant referee flags for offside against Javier Hernandez. Indirect free kick taken by Johnny Heitinga.
Corner taken by Leighton Baines from the right by-line played to the near post, clearance by Danny Welbeck.
Marouane Fellaini takes a shot. David De Gea makes a comfortable save.
Ji-Sung Park takes a shot. Save by Tim Howard.
Seamus Coleman takes a shot. David De Gea makes a save.
The referee gets the game started.
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