Robin van Persie struck a brilliant winner for the Gunners after 52 minutes.
Vito Mannone made several important saves as Fulham created a number of chances.
However, it was Arsenal who got the crucial breakthrough on 51 minutes through a fine finish from Dutch striker van Persie.
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Dimitar Berbatov and John O'Shea struck in the second half to give United the win.
United were struggling to make their dominance pay until Ryan Giggs' introduction 10 minutes after the break.
The veteran Welshman provided a tap-in for Berbatov before O'Shea celebrated his 350th appearance by nodding home Giggs' curling free-kick.
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Liverpool ran out emphatic victors against Hull City at Anfield.
Reds striker Fernando Torres scored a brilliant hat-trick to blow away struggling Hull.
The Spain striker scored twice in the first half either side of Geovanni's equaliser and added a third soon after the interval.
Captain Steven Gerrard added a fourth just before the hour and substitute Ryan Babel had two late strikes as Liverpool made it four league wins - including 16 goals - in succession.
For an in-depth match report, click here.Gabriel Agbonlahor gave the visitors the lead after just three minutes but Chris Samba levelled on 24 minutes.
Vincenzo Grella was sent off for Rovers who nicked the three points a minute from time with a penalty by David Dunn.
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Keane has started every Barclays Premier League match this season and, despite Peter Crouch's midweek treble at Preston North End, he was given the nod again.
The Republic of Ireland striker scored from the penalty spot before splitting Burnley's defence in the build up to Jermaine Jenas' second.
Two cool finishes in the second half meant the 29-year-old was taking the ball home with him, before his fourth squeezed through the legs of goalkeeper Brian Jensen.
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Carlo Ancelotti suffered his first defeat as Chelsea manager with Wigan Athletic chalking up a shock Barclays Premier League victory at the DW Stadium.
After eight straight wins in all competitions the Italian manager saw his 10-man team come unstuck and have goalkeeper Petr Cech sent off.
Titus Bramble put Wigan ahead only for Didier Drogba to pull Chelsea level. The home side regained the lead through a penalty from Hugo Rodallega and Paul Scharner made it safe in stoppage-time.
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Portsmouth suffered a seventh successive Barclays Premier League defeat as they were beaten by Everton.
Louis Saha scored the only goal of the match at Fratton Park three minutes before half-time to give the Merseysiders their first away points of the season.
The Frenchman brought down a long, high ball into the box - beating Younes Kaboul in the air - and pounced, directing a clinical finish past James.
Both sides had an effort cleared off the line in the second half with Pompey also hitting the bar with a header by Aaron Mokoena. Kevin-Prince Boateng was denied by Tim Howard's fine save in stoppage-time.
The home side piled on the pressure late on but could not prevent a Premier League record seventh straight reverse from the start of a campaign. It is only the fourth time in top-flight history that a side has opened with seven defeats.
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