Gervinho had seen his clinical first-half strike from Aaron Ramsey’s pass cancelled out by a close-range goal from former Spurs striker Peter Crouch.
But Van Persie swooped just six minutes after coming on as a second-half substitute to convert Gervinho’s cross.
The Dutchman added a third goal when he diverted a shot past Asmir Begovic after another Gervinho pass.
The win sees Arsenal leapfrog Stoke to go seventh in the Premier League table and maintain a winning sequence at the Emirates Stadium that is now four league games.
It was somewhat harsh, though, on the visitors, who had defended with strength and in numbers and looked good value for a point – until the introduction of van Persie.
Gervinho had already had a sight of goal when he drifted in behind the visiting defence only for Begovic to collect the ball after he miscontrolled it with his chest.
But the Ivorian gave Stoke’s keeper no chance midway through the half, collecting Ramsey’s clever lofted pass and keeping a cool head to slot home his second goal in Arsenal colours.
The strike should have been the signal for the home side to get on the front foot but the opposite happened, with Stoke gaining more possession and being rewarded for their efforts with an equaliser.
Arsenal players protested the free-kick awarded against Lauren Koscielny for a foul on Peter Crouch but they failed to defend it properly, losing three 50-50s in their own box as the visitors restored parity.
First Ryan Shawcross peeled off at the back post to return the ball across goal, where it was flicked on by Matthew Upson to Jon Walters, who nodded it goalwards to former Tottenham striker Peter Crouch to sidefoot home from three yards.
Van Persie was introduced in the second half after being left out of the starting line-up, and made an almost instant impression, sidefooting home Gervinho’s right-wing cross just six minutes after coming on.
The duo repeated the trick eight minutes from time, with Gervinho advancing down the left wing this time and firing a pass into van Persie, with Begovic failing to keep out the Dutchman’s shot.