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.
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.
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:
"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."