In a match where the defenders forgot to show up and most attackers forgot their shooting boots, the Gunners relinquished the lead on two occasions after fighting back from a goal down to see their hopes of finishing third slightly fade away.
Yossi Benayoun put the home side ahead inside two minutes when he curled brilliantly into John Ruddy's top left corner from Tomas Rosicky's pass and it seemed like it would just be another day at the office for Arsene Wenger's men.
But Paul Lambert's charges, as they have done all season, had other ideas and were able to level ten minutes later when Wes Hoolahan got his side-footed attempt from Kyle Naughton's low cross on target with Wojciech Szczesny spilling the ball into the back of his own net.
15 minutes later and the Canaries were ahead when Grant Holt received a ball from Hoolahan at the top of the area to see his attempt deflect off Kieran Gibbs and over Szczesny for his 14th of the season.
Arsenal were rattled as Norwich continued to attack hard and the visitors should have had a penalty late in the first half when defender Russell Martin was pulled down by Laurent Koscielny but referee Anthony Taylor waved it away. The French centre-back then produced a huge tackle to deny Holt in stoppage time.
Simeon Jackson could have had Norwich two goals ahead when he made a barnstorming run from the middle of the pitch seven minutes into the second half but put his shot straight at Szczesny before a number of chances were created by both sides with Ruddy saving a van Persie right-footer and Hoolahan putting his volley inches wide after a great one-two with Holt.
With just under 20 minutes remaining, Arsenal skipper van Persie gave his side a lifeline when he made it 2-2 with a clinical side-footed volley from a deft Alex Song ball eight minutes before the Dutchman found himself on the 30-goal milestone when he pounced on a deflection to hammer past Ruddy with his right foot to give the North Londoners the lead.
With five minutes of regular time remaining, Norwich substitute Steve Morison equalised in fine fashion when he got on the end of Jonny Howson's neat ball to finish sublimely past Szczesny from the right side of the area.
Arsenal were doing all the attacking late in the piece and should have had a penalty of their own when van Persie was clear on goal for a tap-in but got a nudge in the back from Naughton with no decision going the way of the home team.
A plethora of injury-time chances were created by the Gunners with Song's audacious backheel being kept out by Ruddy before the Canaries gloveman also thwarted Alex Oxlade-Chamerblain, Thomas Vermaelen and a Gibbs header to make it eight saves for the game in a heroic performance at the death.
The draw dented Arsenal's automatic Champions League hopes with Tottenham and Newcastle United just two points behind them whilst Norwich remained 14th with another big result in North London after beating Spurs 2-1 last month.
Emirates Stadium, London