The Dutchman recorded his 30th and 31st goals of 2011 to leave his side unbeaten in eight games after Steve Morison's early strike for the Canaries threatened to cause an upset
Norwich took a surprise lead early on as Steve Morison took advantage of a poor mistake by Per Mertesacker, but Van Persie's double ensured a happy afternoon for the Gunners.
A sun-drenched pitch saw the visitors make a lively start to the game, with the pace of Theo Walcott and Gervinho in particular causing the Canaries problems in the wide areas.
Van Persie was also looking menacing in the early stages, dragging his 20-yard effort just inches past John Ruddy’s far post.
With just 10 minutes gone, the visitors were left wondering how on Earth they hadn’t taken the lead.
Following persistent pressing from Gervinho, Walcott received the ball 15 yards from goal. The winger cut inside onto his left foot, before curling a brilliant effort towards the far corner, only for Russell Martin to defy the England man with a superb hooked clearance.
And as so often is the case, Arsenal were left to rue their missed chances as Norwich took the lead just minutes later.
Summer signing Mertesacker was hassled by Morison on the edge of the area, and the forward’s pressure paid off as the German centre-back crumbled, leaving the Welshman one-on-one with Szczesny and able to coolly slot home.
The goal didn’t deter the visitors, though, as they continued in the same attacking fashion. Next it was Gervinho who was thwarted, as his low effort was blocked by John Ruddy.
They did, however, eventually get the goal their play deserved.
Unsurprisingly it was Walcott who created the goal, driving to the by-line before pulling the ball back for Van Persie, who was left with the simple task of restoring parity for his side. The goal was the Dutchman’s 14th in his last 17 games.
Just before the hour, Arsene Wenger’s men should have taken lead for the first time in the game. As so often, Van Persie was the lynchpin in the attack, displaying some brilliant footwork before threading the ball through for Gervinho.
The Ivorian sidestepped the onrushing Ruddy, and with the hard work done and an empty net gaping, he contrived to fire the ball at the diving keeper.
After seeing his team-mates squander multiple chances, Van Persie took matters into his own hands and gave his side the lead.
Martin, who hadn’t put a foot wrong all game, was caught in possession in the centre circle by Aaron Ramsey. The ball fell for Alex Song, who slid Van Persie through, and the Dutchman did the rest, superbly chipping the ball over Ruddy with his weaker right foot.
From then on it was an Arsenal masterclass, frustrating the home side with some excellent keep-ball.
The Canaries continued to search for an equaliser, but the late onslaught, which the Carrow Road faithful have become so accustomed to, did not materialise.
The win leaves Norwich in ninth, while Arsenal move up a place to fifth, level on points with London rivals Chelsea and Tottenham.