Samir Nasri and Abou Diaby also scored for the Gunners at Emirates Stadium.
Arsenal lead Group H with 10 points. Despite the loss, AZ still has a chance to qualify from its last two matches because second-place Olympiakos lost 2-0 to Standard Liege.
The Dutch team has two points, Olympiakos has six and Liege has five.
“It was a brilliant performance because we dominated our subject technically, tactically, and overall we never felt under threat,” Arsenal manager Arsene Wenger said. “We played with the needed confidence. Overall, I’m very happy.”
Fabregas’ weak shot in the 25th minute should have been stopped by goalkeeper Sergio Romero but beat him at the near post. Romero seemed surprised by Fabregas’ shot and was slow to dive to his right, barely getting his fingertips on the ball.
“The first goal our goalkeeper can do something more with that shot,” AZ coach Ronald Koeman said. “But he (Fabregas) is a fabulous player.”
Fabregas’ first goal spurred AZ into action and the Dutch side pressed the ball further up the pitch. Maarten Martens surged forward in the 30th minute with a good run, but his shot was easily caught by Arsenal goalkeeper Manuel Almunia – his first save of the night.
Two minutes later, midfielder Brett Holman fired a shot from distance but the ball swerved wide.
Arsenal’s second goal came just before halftime when Nasri collected a throughball from Andrei Arshavin. The France midfielder shrugged off a defender and curled a low shot past Romero, who again got his fingers to the ball.
It was Nasri’s second appearance in all competitions this season since returning from a broken leg he sustained in preseason training.
Fabregas scored his second in the 52nd minute after AZ conceded possession and Arshavin’s throughball found the Spain international, who curled a shot over Romero.
AZ nearly scored five minutes later when Graziano Pelle latched onto a deep cross and his shot from close range struck the crossbar.
A quick counterattack led to Arsenal’s fourth goal, after Eduardo da Silva’s backheel flick found Arshavin. The Russia international blazed down the right before squaring a pass to Diaby on the left who fired a shot past Romero.
AZ pulled one back in the 82nd minute on a counterattack, with second-half substitute Jeremain Lens firing a low shot past Almunia.
“In the first 25 minutes, I thought the organization of the team was quite good,” Koeman said. “But in front of goal we weren’t able to do much.
It was the first time that Koeman lost to Arsenal in six matches dating back to 2003. In 2007, Koeman was manager of PSV Eindhoven when it eliminated Arsenal from the first knockout round of the Champions League.
“I don’t think we’ll qualify for the Champions League, but if we win two more games it’s possible – it’s not impossible,” Koeman said. “It’s very clear in the next game against Olympiakos we have to win.”