Arsenal have 32 points with Chelsea, who have taken just five points from their last six league games, third on 30, one behind Manchester United whose game at Blackpool was frozen off.
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Manchester City are fourth on 29 after they ended Bolton Wanderers' good run with a 1-0 Eastlands victory courtesy of an early Carlos Tevez goal.
Fifth-placed Tottenham Hotspur lost ground when they could only draw 1-1 at Birmingham City while Blackburn Rovers, thrashed 7-1 by Manchester United last week, bounced back impressively by beating Wolverhampton Wanderers 3-0.
Arsenal, whose players and staff all donated their day's wages to charity, were all over Fulham in the first half-hour but had only a cool finish by Nasri to show for it.
Perhaps the easy ride their defenders were enjoying contributed to the comedy of errors that opened the way for Fulham's equaliser as Laurent Koscielny and Sebastian Squillaci clashed heads and stopped playing, allowing Clint Dempsey to set up Diomansy Kamara for a nicely curled finish.
It took a while for Arsenal to regain control and they still managed to waste several good opportunities before Nasri won it with a superb goal after 75 minutes when he neatly rounded a defender and the goalkeeper before scoring his 11th of the season from a tight angle.
"We had to work very hard because Fulham had a great game as well but my players showed fantastic spirit and quality," Arsenal boss Arsene Wenger told reporters.
"We went for it in the second half. Overall, we deserved it. Credit to Nasri, his goals were outstanding. He showed skill, intelligence and calm. He's just been voted French player of the year and it underlines how important confidence is."
STUTTERING CHAMPIONS
Chelsea had John Terry and Michael Essien back from their injury absences to stiffen both their spine and their resolve but, as so often in recent weeks, they looked anything but champions and ended up hanging on desperately.
Chelsea, who hit the bar early through Terry, were on top but struggling with their final pass until Everton captain Phil Neville provided one for them with a badly under-hit backpass three minutes before half-time.
Striker Nicolas Anelka was onto it in a flash only to be brought down by goalkeeper Tim Howard, with Didier Drogba scoring the penalty.
Everton, who have been suffering their own poor run in recent weeks culminating in a 4-1 home defeat by West Bromwich Albion last week, refused to buckle and took the game to the hosts in he second half.
Jack Rodwell hit the post with a header from a Leighton Baines cross but another surging run past three defenders by the full-back set up the equaliser three minutes from time when his cross led to Jermaine Beckford heading past Petr Cech.
Chelsea were booed off by their own fans and now face a potentially decisive month as they play Tottenham Hotspur, Manchester United and Arsenal.
