Theo Walcott gave the hosts the lead inside the second minute as Arsenal raced out of the blocks but the Gunners were pegged back at the end of the half courtesy of a van Persie penalty.
After a week's worth of speculation, van Persie started on his return to the Emirates while Lukas Podolski also made a rare start as Arsenal's central striker with Olivier Giroud missing with suspension.
The Gunners got off to the perfect start in front of their red hot home support when Walcott beat the offside trap and calmly found a way past David de Gea from a tight angle inside the second minute.
Santi Cazorla then stung the hands of the United keeper with a shot from range as Arsenal maintained their early momentum.
Van Persie's first real involvement in the game was a miscued shot which sailed high over the bar just after the 10 minute mark, much to the delight of the Gunners' supporters. Those fans then collectively held their breath moments later when the Dutchman broke down the left flank before crossing for Phil Jones who failed to find the target with a header after surging into the box.
Wenger's men lost some of that early impetus as the first half went on but were sparked back into life when Jonny Evans brought down Walcott just outside the area.
The free kick was wasted but moments later Arsenal had a glorious chance to double its lead on the counterattack. Rosicky elected to pass to Podolski, rather than the unmarked Walcott, but the German could only fire his shot from a tight angle straight at De Gea.
Arsenal then had Wojciech Szczesny to thank for keeping it ahead just before the break. The Poland international made himself big to keep out van Persie's header from point-blank range after a swift United counterattack.
But moments later van Persie made no mistake, this time from the spot, after Sagna's rash challenge on the Dutchman gave United a penalty.
The Gunners made a bright start after the break and could have gone ahead early on but Tomas Rosicky's prod with the outside of his boot whistled just past the De Gea's far post.
Sir Alex's side soon edged back into the contest and dictated the game for a period of 15 minutes. Van Persie over-hit a low cross past the onrushing Antonio Valencia before Wayne Rooney wasted a good chance with a header straight at Szczesny from close range.
Arsenal rallied for a winner in a end-to-end finale to match with the visitors still a threat on the break but neither side could fashion the clear-cut chance needed to win the game.
Cazorla came close with a powerful right-footed shot which was palmed away by De Gea with minutes left on the clock before Ryan Giggs wasted a great opportunity when found totally unmarked on the break by Rooney.