The Holland striker, who used his programme notes to declare his full commitment to Arsenal, put the hosts ahead after just 29 seconds, but the sides went in level at half-time thanks to an excellent free-kick from former Gunner Sebastian Larsson.
The game appeared destined for a draw as Arsenal proceeded to squander several good second-half chances, but Van Persie saved the Londoners' blushes with a wonderful curling free-kick nine minutes from time.
The win gave Arsenal the perfect tonic after their north London derby defeat two weeks ago, but the creaky manner in which they snuck past struggling Sunderland will do little to silence the doubters who claim the Gunners are a spent force.
Under-pressure Black Cats boss Steve Bruce, meanwhile, will rue his side's failure to convert a number of first-half opportunities and they now sit outside the drop zone only on goal difference.
Van Persie was also denied an early contender for goal of the season by the woodwork when he spun his marker on the edge of the box in the 13th minute and clipped a delightful chip that sailed over Mignolet but bounced out off the post, with Gervinho just missing out on the tap-in.