The Gunners were terrific in the first half and were rewarded when Cazorla struck on 35 minutes following a flowing move.
Jack Wilshere had to be replaced just after the interval due to injury and Carl Jenkinson's dismissal allowed Sunderland to start dominating, but Wojciech Szczesny and Bacary Sagna led a magnificent rearguard effort as Arsenal boosted their hopes of a top-four finish.
Sunderland battled hard in the early stages to try and stop Arsenal finding any fluency, but it did not take Wilshere and Cazorla long to start dictating play with their expert touch and vibrant movement.
Simon Mignolet had to pull off a series of saves to keep Sunderland on level terms, with one particularly athletic dive to his left to parry away Aaron Ramsey's strike from 20 yards.
The Gunners were not to be denied, though, and Cazorla opened the scoring with 10 minutes remaining of the first half when he arrowed a low shot through a crowded area after being fed by Theo Walcott.
Ramsey should have made it 2-0 just before the interval after being played in by Olivier Giroud, but Mignolet rushed off his line to make another wonderful stop from point-blank range.
Wilshere's 50th-minute departure after he was fouled by Alfred N'Diaye removed one threat for the home side, but added Abou Diaby's physicality to the mix.
Sunderland should have been back on terms just before the hour mark when Ramsey was caught in possession on the halfway line to allow Stephane Sessegnon to race upfield and lay the ball into Steven 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.
Sunderland received a boost in the 62nd minute when Jenkinson, who had been booked for a ninth-minute foul on Jack Colback, scythed Sessegnon down and received a second yellow card.
Black Cats boss Martin O'Neill went for broke with 22 minutes remaining when he sent on Danny Graham, but Arsenal continued to pose a threat as Diaby and Cazorla went close and Walcott hit the post after chipping an effort over the advancing Mignolet.
Per Mertesacker's 78th-minute blunder handed Fletcher another golden opportunity, but Szczesny was equal to the task, and the Poland international produced another excellent save five minutes from time to keep out the Scot's goalbound header.
Defender Titus Bramble could have snatched a point seconds later, but he could not direct his close-range effort at goal and Szczesny kept out Adam Johnson's injury-time effort with another good save.