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.
Arsene Wenger stuck with the Arsenal starting XI that won 2-1 at Aston Villa on Monday, while Rene Meulensteen made three changes from the team that beat Norwich City 3-0 in the FA Cup on Tuesday, handing a Premier League debut to defender Dan Burn.
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.
The hosts continued to dominate possession, but were struggling to carve out any meaningful goalscoring opportunities as the Fulham back line held firm.
And they were almost punished for that lack of potency in the 26th minute when former Arsenal trainee Steve Sidwell, who scored in his last three appearances for Fulham prior to the trip to the Emirates, forced Wojciech Szczesny into a crucial save by sending a powerful 20-yard half volley goalwards.
Maarten Stekelenburg had to be equally alert at the other end seven minutes before the break when a free-kick ricocheted into the path of Bacary Sagna, who saw a low drive well stopped by the Dutchman.
Fulham came out fighting at the start of the second period, as Burn had a header blocked by Laurent Koscielny after Arsenal failed to clear their lines from a corner.
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.
And Cazorla doubled the advantage five minutes later, firing a low 20-yard strike just out of the reach of the outstretched left arm of Stekelenburg.
Stekelenburg denied Arsenal a third 15 minutes from time when he pushed a Lukas Podolski thunderbolt - destined for the top-left corner - onto the post.
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.
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