Overshadowed by Robin van Persie this season, Rooney showed his predatory instincts after 79 minutes at Craven Cottage when he punished a defensive error to grab an unconvincing win that turned up the heat on Manchester City who play on Sunday.
Faltering European champions Chelsea suffered their second late collapse this week, going down 3-2 at Newcastle United having led 2-1 with goals from Frank Lampard and Juan Mata.
Newcastle's new signing Moussa Sissoko equalised and then capped a superb home debut with a last-minute winner to increase the pressure on Chelsea's interim manager Rafa Benitez whose side conceded twice late on in a midweek draw at Reading.
Third-placed Chelsea have won only once in six games in all competitions and alarm bells are ringing about their hopes of sealing a top-four finish that brings a Champions League spot.
With Tottenham Hotspur playing on Sunday, Everton had a chance to move above them into fourth but had to fight back for a 3-3 draw at home to struggling Aston Villa, who led 3-1 before Marouane Fellaini's double spared the home side's blushes.
Lukas Podolski scored in the 78th minute to hand Arsenal a 1-0 win against visiting Stoke City, a result that helped the Gunners close in on the Champions League qualification places.
Manchester United have 62 points from 25 games with City, who host Liverpool on Sunday, on 52 and Chelsea on 46.
Spurs, who visit West Bromwich Albion on Sunday, have 42 points along with Everton while Arsenal are on 41.
Despite a yawning gap at the top, Rooney said the title was far from being in the bag.
"We're in a good position but we're not getting carried away," he told ESPN. "Let's see where we are with three or four games to go."
United were not at their best and had an escape in the first half when Fulham's Brian Ruiz hit the inside of the post with a volley, although they carved out plenty of chances and Patrice Evra and Rooney were both denied by the woodwork.
A scrappy second half came to life late on when Fulham defender Philippe Senderos misjudged a high ball and Rooney raced clear to put United in front.
Van Persie was needed at the other end to clear a Senderos header off the line as United hung on for their 20th league victory of the season.
"I think it was a really determined performance and it wasn't easy because Fulham played their part. 1-0 doesn't do the game justice," said United manager Alex Ferguson.
Any thoughts of putting some pressure on the Manchester clubs in the title race are long gone for Chelsea who were left reeling after another late horror show.
Benitez said his side should have been awarded a penalty and the hosts reduced to 10 men after striker Demba Ba, signed from Newcastle last month, was left bloodied by a clash with Fabricio Coloccini after trying to turn in a rebound before halftime.
"It was a penalty and a red card," Benitez told Sky Sports, saying Ba, who was replaced by Fernando Torres, had suffered a broken nose in the incident.
"If it had been in the middle of the park it would have been a free kick and maybe a yellow but in the area it's a penalty and red because it was a clear chance."
Worryingly for Chelsea, they appear to be running out of steam and conceding late goals just as the season reaches a critical point in the battle for a top-four finish.
It was the case again on Saturday as Sissoko turned the match on its head after Chelsea had seized control.
Jonas Gutierrez gave Newcastle the lead after 41 minutes with a header but Chelsea replied after the break with Lampard's fierce shot and a curler from Mata.
Former Toulouse midfielder Sissoko, one of several French signings for Newcastle during January, tapped in a 68th minute equaliser after Chelsea keeper Petr Cech parried Papiss Cisse's shot.
'PAYING US BACK'
Sissoko then crashed home a 90th-minute winner after a cut-back from Davide Santon.
"I remember watching him for Toulouse 14 months ago, and I thought he was the one for us," Newcastle boss Alan Pardew said.
"We accelerated (the deal) because he was going to come in the summer and he's paying us back handsomely."
David Moyes's Everton seemed to be slipping to only their fourth league defeat of the season after Christian Benteke's double helped Villa into a 3-1 lead at Goodison Park.
However, Fellaini took the game by the scruff of the neck, scoring with a low shot 20 minutes from time and then heading an equaliser in the third minute of stoppage time.
At the bottom, a missed penalty by Adel Taarabt cost relegation-threatened Queens Park Rangers victory against struggling Norwich City as they drew 0-0 at Loftus Road.
QPR's fourth league draw in a row means they are now six points from safety after Reading beat Sunderland 2-1.
Southampton drew 2-2 at Wigan Athletic, meaning none of the bottom six sides before kickoff lost.
West Ham United beat Swansea City 1-0 thanks to a goal from on-loan Liverpool striker Andy Carroll.
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