© PA Photos
The Gunners, who will move four points clear of fifth-placed Tottenham Hotspur if they win today, broke the deadlock after just three minutes courtesy of Theo Walcott.
Van Persie conceded possession on the halfway line, allowing the hosts to pounce with a fluid counter-attack.
Tomas Rosicky slid the ball through to Walcott, who was marginally offside, but the flag stayed down and the England forward finished well from eight yards.
United came close to levelling on the quarter-hour mark, but Phil Jones headed just wide following a brilliant run-and-cross by Van Persie.
Van Persie was then thwarted by an incredible point-blank save from Wojciech Szczesny, who blocked the Dutchman's close-range header with his face.
However, Van Persie was not to be denied in the final minute of the first half.
Szczesny dived the correct way, but Van Persie's effort was too good.
United have not received a red card in domestic competitions all season, but they will have to improve their discipline in the second half if that run is to continue.