After a goalless opening period in which Arsenal struggled to create many chances, the Spaniard broke the deadlock shortly before the hour mark, combining well with Olivier Giroud and Jack Wilshere.
And his tally was doubled five minutes later when he buried a left-footed 20-yard effort in the bottom right-hand corner of the Fulham net.
The result maintains Fulham's run of having never won at Arsenal, who remain one point clear of Manchester City at the summit of the English top flight.
Mesut Ozil almost had Arsenal in front in the fourth minute as Wenger's men made a bright start, seeing an effort blocked in the six-yard box by Brede Hangeland following a neat exchange of passes with Wilshere in the penalty area.
But the hosts weathered that early storm and Serge Gnabry drew another save from Stekelenburg with a curled effort from the edge of the penalty area, before Arsenal squandered numerous opportunities in a goalmouth scramble.
Fulham's resolve was finally broken in the 57th minute when Cazorla picked up the ball on the left wing, offloaded it to Giroud, before ghosting into the penalty area and sliding into the bottom right-hand corner from a Wilshere pass.
Darren Bent found himself clean through on an open goal in the closing stages, but the Fulham striker fired wide of the target as Koscielny slid in to make a last-ditch challenge, before the visitors failed to capitalise on Szczesny's spillage from Pajtim Kasami's stoppage-time free-kick.
