Arsenal raced into an early lead through Theo Walcott but Van Persie levelled the match before the break from the penalty spot.
Both sides had chances to win it in the second half but had to settle for the draw.
Here, Sports Mole picks the bones out of a feisty tussle in North London.
On target 13
On target 5
Was the result fair?
Both managers would have felt hard done by had they left with nothing so a draw was a fair result. Arsenal were out of the blocks extremely quickly but couldn't open up a two-goal lead and soon United were the dominant force. The Red Devils were on top at the start of the second half as well, but Arsenal ended the game the stronger and overall a point apiece is about right.
Arsene Wenger's men were blisteringly brilliant in the early stages and deservedly took the lead very early on. Walcott may have been slightly offside but it was just what the Gunners deserved. They couldn't quite find a second in the first half an hour and then shot themselves in the foot by conceding a needless penalty that former frontman Van Persie made no mistake in converting. In the second half they didn't have quite the same zest but still threw bodies forward, almost conceding on a number of occasions on the counter-attack.
Sir Alex Ferguson's side were given a guard of honour by the red shirts for their entry but they started like they had been out partying every night since Monday's title-winning performance. It wasn't until after half an hour that they finally started to play and the equaliser was coming after a pair of misses from Van Persie and Phil Jones. In the second half they definitely tried to inflict damage on the counter but they couldn't quite find the winner as the defence stifled Arsenal down the other end.
Sports Mole's man of the match
Phil Jones: Tough to pick out one man who shone out this afternoon as it was a frenetic encounter from start to finish. United's young defender-cum-midfielder Jones is our winner today, though, for his energetic performance that saw him influence proceedings at both ends of the pitch.
This award is going to Bacary Sagna for his careless tackle on Van Persie that led to the equalising penalty. The Frenchman presented his former teammate with the ball and tried to atone for his error but could only hack down the Dutchman.
It was a tough game for Phil Dowd to officiate with plenty of tasty challenges that resulted in a total of eight bookings - five for United, three for Arsenal. The one big contentious call came when Antonio Valencia handled the ball and Dowd correctly ruled for a free kick despite many calls for a penalty. Walcott's goal was offside but that is Dowd's assistant's job to spot.
Arsenal: Next up for the Gunners is a trip to now-relegated Queens Park Rangers on Saturday.
United: Another tough game is next up for Ferguson's side as they welcome Chelsea to Old Trafford in a week's time.