The Holland striker put us 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 we proceeded to squander several good second-half chances, but Van Persie saved our blushes with a wonderful curling free-kick nine minutes from time.
The win gave us the perfect tonic after the north London derby defeat two weeks ago, but the creaky manner in which we snuck past struggling Sunderland will do little to silence the doubters who claim we're 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.
Keen to make up for the loss at White Hart Lane, we flew out of the blocks and took an early lead.
Some fans had still not taken their seats by the time that Gervinho got on the end of Tomas Rosicky's lobbed pass and squared to Van Persie, who took one touch before drilling low past Simon Mignolet from 18 yards to record the fastest top-flight goal of the season so far.
The strike stunned the Black Cats and they remained on the back foot for the rest of the opening half hour.
This was the Arsenal of old. Confident in possession and sturdy at the back, the lads pushed for a second as the away side struggled to get a touch on the ball.
Van Persie was then denied an early contender for goal of the season by the woodwork.
With 13 minutes on the clock, the Dutchman spun his marker on the edge of the box and clipped a delightful chip that sailed over Mignolet but bounced out off the post and Gervinho just missed out on the tap-in.
Full of confidence, Van Persie than flashed a powerful 20-yard drive just wide.
Gervinho was also in his pomp. The Ivory Coast man cut inside from the left flank with a clever turn and crashed a fierce shot just over.
Wojciech Szczesny then went walkabout, handing Sessegnon a glorious chance to equalise, but the winger's cross was cut out by Alex Song as the Arsenal goal lay unguarded.
Our confidence was knocked and Sunderland shocked the home crowd with a stunning equaliser from Larsson.
Mikel Arteta's handball gave the Black Cats a free-kick 20 yards out and Larsson whipped a delightful curling shot over the wall and past a sprawling Szczesny to make it 1-1.
The strike was so good that Rosicky had already started walking to the halfway line without turning his back to see if the shot had gone in.
Szczesny denied Sunderland the lead moments later when he blocked Lee Cattermole's header from six yards after an error from Carl Jenkinson allowed Sessegnon to break down the left.
The game became scrappy at the start of the second half.
David Vaughan earned the first booking of the game with a cynical two-footed sliding tackle on Rosicky and he was soon followed in to the book by Larsson, who tugged Van Persie's shirt.
Tensions mounted as Alex Song was booked for a foul on Sessegnon and then John O'Shea upended Rosicky on the edge of the box, giving us a free-kick which Arteta duly ploughed in to the stands.
Brown was lucky to escape with a booking after clattering into Walcott in a similar position minutes later. The England winger picked himself up to take the free-kick but the home fans groaned collectively as, again, the set-piece flew well over.
Howard Webb produced his fifth yellow card of the game after Laurent Koscielny body-checked Sessegnon before Mignolet kept Van Persie's near-post effort out with his feet as we increased the pressure on the visitors.
Arsene Wenger looked despondent on the bench as Alex Song then broke into the box and inexplicably passed to a Sunderland defender with time running out.
A moment of brilliance from substitute Andrey Arshavin almost gave us the lead with just under 20 minutes left.
The Russian danced past two Sunderland defenders into the box and nutmegged Michael Turner with a shot that went a yard wide of Mignolet's goal.
Brown's clumsy challenge on Van Persie gave us another free-kick on the edge of the box with nine minutes left. The Holland international succeeded where his team-mates had failed before him by curling a brilliant set-piece into the top corner and then removing his shirt in a jubilant celebration.
Ji Dong-won was then played in to the box by Connor Wickham and slotted past Szczesny but his effort was rightly ruled out for offside.
Sunderland pushed for a second equaliser towards the end, but it never came and Wenger's side took the spoils after another nervy afternoon.
Source: DSG