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They took the lead after just three minutes when Van Persie struck from outside the box to score against his former club.
Arsenal began to enjoy a period of possession as they settled following the opening goal, but they struggled to create any chances of note.
United continued to create chances, however, with Wayne Rooney and Van Persie both coming close either side of the 30-minute mark in the match.
Rooney had the chance to extend United's lead from the penalty spot late in the first period, but he fired wide of the post after Santi Cazorla had handled in the area.