Van Persie chalked up his 30th and 31st goals of 2011 as the Gunners bounced back from a 16th-minute goal by Steve Morison to triumph at Carrow Road and move above Liverpool into sixth place in the Premier League table.
Mertesacker was robbed by Morison on the edge of the area as the Wales striker smashed the Canaries ahead against the run of play.
However, Arsenal's pressure finally told 10 minutes later when van Persie slotted home after good work by Theo Walcott down the right.
And the Dutchman made it 10 goals in his last five league games when he chipped a shot over John Ruddy after a fine through-ball by Alex Song just before the hour mark.
Fired by van Persie's goals, Arsene Wenger's side have now chalked up five straight league victories to close to within striking distance of the top four, and are now transformed from the shambles of the start of the campaign.
Wenger had named a strong starting XI in Norfolk, resisting the temptation to rest some of his key men after the international break.
The Gunners signalled their early intent as Walcott, who played twice for England at Wembley, got away down the right and cut the ball back for van Persie, but for once he failed to connect properly and dragged the shot wide.
After Norwich ventured forwards, the visitors were soon back on the offensive, van Persie driving a low shot across the face of Ruddy's goal after latching onto Andre Santos' pass.
After 12 minutes, it took an acrobatic clearance by Russell Martin to deflect Walcott's curling left-foot effort wide just when the ball was destined for the bottom corner following a quick counter-attack by Gervinho.
The Gunners continued to pour forwards as Norwich were stretched down both flanks.
However, yet again Arsenal were undone by a mistake at the back.
Mertesacker failed to shield what should have been a routine ball through to the keeper, which allowed Morison to outmuscle the 6ft 4in German before slipping a shot past Wojciech Szczesny to give the Canaries the lead on 16 minutes against the run of play.
After such a period of dominance, Arsenal were shell-shocked, but soon went back on the offensive.
Martin was again in the right place to knock van Persie's looping header clear.
Norwich keeper Ruddy and the Scotland defender then combined to hack away Gervinho's angled drive after the Ivory Coast wideman had cut in from the left.
The equaliser eventually arrived on 26 minutes when Walcott skipped past Marc Tierney to cut the ball back through the six-yard box, where Gervinho failed to connect with a reverse flick, which fortunately presented van Persie with a tap-in for his ninth goal in five Premier League games.
To their credit, Norwich were not overrun, and continued to make progress in the Arsenal final third, with Morison battling well supported by Anthony Pilkington.
However, more often than not, the home side opted for the wrong choice of pass with men over.
Walcott switched to the left and continued to torment the Norwich backline, but Arsenal could not find a way through ahead of the interval.
Norwich replaced David Fox with winger Elliott Bennett, a £1million summer signing from Brighton, for the second half.
The change almost paid dividend when Bennett's ball from the right flew across the Arsenal area.
Thomas Vermaelen then scampered forwards and his cut-back was glanced goalwards by Gervinho, but without any real power to trouble Ruddy.
Walcott was sent clear away down the right, and his near post cut-back was slotted into the sidenetting by the on-rushing Aaron Ramsey.
Norwich were cut open again when a sweeping Arsenal counter saw van Persie slip Gervinho through the centre, but Ruddy was out quickly to block before the loose ball was then hacked away.
On the hour, the Dutchman showed him how it should be done when referee Phil Dowd played a fine advantage after Ramsey had been tripped, which allowed Alex Song to send van Persie away in the right channel.
The Gunners skipper took one touch before clipping a superb, angled finish over Ruddy.
To their credit, Norwich quickly tried to get themselves back into the game, with Bennett's low cross only needing a touch from Pilkington at the near post.
Paul Lambert's men have shown great deal of character this season, earning a 1-1 draw at Liverpool, and refused to accept defeat, with substitute Grant Holt giving Mertesacker a real test.
However, the Norwich club captain was booked for taking a tumble in the box as Arsenal kept possession well during the closing stages to close out a fifth straight Premier League win as their resurgence continued.