Wayne Rooney dragged a penalty wide in first-half injury time for United, but the England striker crossed for Patrice Evra to head in the clinching goal in the 67th minute.
Arsenal, which played the final 21 minutes with 10 men after Jack Wilshere was sent off for two bookings, grabbed a consolation goal through Santi Cazorla with the final kick of the game.
United is already nine points clear of Arsenal after just 10 games and the team is being driven on by Van Persie, who left the Gunners after eight seasons for 24 million pounds (then $38 million) in search of silverware.
He was received warmly by the visitors — Arsenal left back Andre Santos even asked the Dutchman to swap shirts at half time — and didn't celebrate his goal that came after Thomas Vermaelen failed to clear Rafael da Silva's cross.
"It's the first time I've ever played against a former club," Van Persie said. "So it's a special day but it's about the game and we won it.
"I played there for eight years, I had a fantastic time there and I respect the fans, manager, players and the whole club so that is why I didn't celebrate."
Van Persie's low, right-foot finish was one of a string of good early chances created by United against a porous Arsenal defense, with thoughts inevitably turning to the 8-2 scoreline in the corresponding fixture last season.
However, Arsenal slowly got back into the match and, buoyed by Rooney's penalty miss after Cazorla was adjudged to have handled Ashley Young's cross, probed for an equalizer in a sprightly start to the second half.
Olivier Giroud struck a close-range shot against the post on the turn and Wilshere — playing his second match back after serious injury — helped the visitors dominate the midfield area for a spell.
But Evra's header from Rooney's inviting cross all but secured victory, moments after Van Persie had seen a golden chance turned aside by goalkeeper Vito Mannone.
When the tiring Wilshere received a second booking, for a studs-first foul on Evra as he stretched for the ball, it turned into a damage-limitation exercise for Arsenal.
Anderson, a substitute for the yellow-carded Tom Cleverley, had a goal disallowed for a marginal offside in the 80th and Rooney also wasted a good chance as United coasted toward victory.
Cazorla curled in a fine goal in the last seconds, but that was immediately followed by the final whistle.