The former Gunner headed home from a corner in the first half to give his side the lead, before the hosts held on for an important victory against Arsene Wenger's below-par team.
It was no surprise that Van Persie was Arsenal's nemesis. The striker has now scored in the last five matches between the two teams and his 27th-minute header was responsible for deciding this encounter, inflicting a blow to the momentum Arsene Wenger's men have built up this term.
Neither side was at its best during the much-anticipated fixture and, although Arsenal improved after the break, it lacked cutting edge with Wayne Rooney going closest to scoring in the second half when he shot just wide.
One goal was enough for David Moyes' side, though, which is now unbeaten in nine matches. Arsenal however – which entered the match on the back of wins over Liverpool and Borussia Dortmund – failed to reinstate its five-point lead at the top of the Premier League.
Van Persie headlined Manchester United's four changes from its goalless draw against Real Sociedad on Wednesday, while Michael Carrick and Jonny Evans also came in after overcoming minor calf and back injuries.
A sickness bug forced Arsenal into two changes, Thomas Vermaelen and Mathieu Flamini coming in for Per Mertesacker and Tomas Rosicky, who were not even among the replacements.
Arsenal shaded a cagey opening in which United's Rooney was the only player to test a goalkeeper, albeit with a relatively tame effort from distance.
Van Persie breathed life into the contest, though, converting the first genuine chance of the game from a Rooney corner. The Netherlands star rose above Oliver Giroud and produced a soaring header that crashed into the top-right corner, too well-placed for Kieran Gibbs – who was on the line – to do anything about.
Arsenal was unable to muster a significant response before the break, but was thankful to see goalkeeper Wojciech Szczesny emerge from a nasty clash of heads with Phil Jones in good health.
Manchester United was not so lucky, with defender Nemanja Vidic failing to return after being hit on the head by the hip of his own goalkeeper, David De Gea, on the stroke of halftime.
Arsenal improved after the break but struggled to create clear-cut chances, Mesut Ozil going closest when firing into the side-netting in the 58th minute.
Rooney nearly doubled United's lead on the hour mark after Arsenal's defense failed to clear a clever flick-on header from Shinji Kagawa, but, after feigning to shoot and creating space, the England star tugged his left-footed effort past the right post with Szczesny beaten.
A long-range effort from Gibbs tested De Gea when it took a late van Persie deflection while Chris Smalling failed to take advantage of a teasing cross from the goalscorer at the other end.
There was a late chance for Giroud, but his shot on the turn went wide as United held on for a crucial win that sees them rise to fifth in the table.