The Spaniard struck 10 minutes before the break to give the Gunners a deserved lead as Black Cats keeper Simon Mignolet turned in another fine display to otherwise keep them at bay.
However, the 62nd-minute dismissal of defender Carl Jenkinson, who was only included after Laurent Koscielny injured himself during the warm-up, for a second bookable offence left the Londoners hanging on in front of a crowd of 46,402.
Steven Fletcher had earlier passed up a glorious opportunity to level but, even after Jenkinson’s departure, Arsenal had chances and Theo Walcott was unfortunate to see a 75th-minute effort come back off the post.
However, goalkeeper Wojciech Szczesny spared Per Mertesacker’s blushes three minutes later when he blocked Fletcher’s shot after he had charged down the defender’s attempted clearance and the visitors managed to see the game out amid a frenzied conclusion.
In a promising start, Black Cats midfielder Alfred N’Diaye looped a first-minute header over Szczesny’s crossbar and only seconds more had elapsed when Walcott forced Mignolet to make the first of what proved to be a series of important saves.
Mignolet threw himself to his left to turn away Aaron Ramsey’s stinging 32nd-minute drive but the respite proved only temporary. The visitors finally took the lead three minutes later when Jack Wilshere and Walcott combined to set up Cazorla on the edge of the box and he drilled a low shot past Mignolet.
Wilshere’s 50th-minute departure after he was fouled by N’Diaye removed one threat for the home side but added Abou Diaby’s physicality to the mix. Olivier Giroud pushed a left-foot shot just wide of the far post with 56 minutes gone but Sunderland should have been back on terms two minutes later.
Ramsey was caught in possession on halfway to allow Stephane Sessegnon to race upfield and lay the ball into Fletcher’s path. The Scotland international looked odds-on to take his tally to the season to 11 but uncharacteristically sliced his left-foot effort wide of the post.
However, Sunderland received a boost seconds later when Jenkinson, who had been booked for a ninth-minute foul on Jack Colback, scythed Sessegnon down and received a second yellow card but they could not find an equaliser.
Rangers defender David Bates faces at least six weeks out with ankle ligament damage.