Rashford, 18, scored twice in the first period for the Red Devils, before setting up Ander Herrera for what proved to be winning goal for a youthful home team.
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Man United welcomed goalkeeper David de Gea back from injury, but Chris Smalling and Anthony Martial failed late fitness tests, which allowed Rashford to make his Premier League debut, while Michael Carrick continued at centre-back.
As for Arsenal, there were three changes from the team that lost against Barcelona in the Champions League - Theo Walcott, Danny Welbeck and Gabriel Paulista in for Oliver Giroud, Alex Oxlade-Chamberlain and Per Mertesacker.
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Alexis Sanchez then had a go from the edge of the box in the 20th minute as Arsenal continued to look dangerous in the final third due to their pace and movement, but the Chilean missed the post.
It was Man United that had the lead in the 29th minute, however, when Rashford, on his Premier League debut, stuck the ball in the top corner after Arsenal had failed to clear a cross from Guillermo Varela.
Arsenal were back in the game in the latter stages of the first period, however, when former Man United youngster Welbeck headed an Ozil free kick into the bottom corner after losing Marcos Rojo inside the box.
The Gunners pushed for a leveller in the latter stages of the first half and came close through Sanchez, who had a long-range strike blocked behind, but Man United entered the interval with a 2-1 lead at Old Trafford.
Arsenal dominated possession in the early stages of the second period and the injury bug hit Man United again, with the returning Rojo forced off in the 55th minute, allowing Timothy Fosu-Mensah to make his Premier League debut.
Schneiderin found himself unmarked inside the Arsenal box in the 56th minute, but his strike was blocked during a period where the home side managed to enjoy an extended period of ball possession. Depay was next to test Cech from distance, but once again the Gunners stopper got down to make the save.
Man United had their third in the 65th minute, however, when Rashford collected a through-pass from Mata, turned and then set up Herrera, whose strike from the edge of the box deflected off Laurent Koscielny into the back of the net.
Arsenal hit back just four minutes later through Ozil, who registered from inside the box after De Gea had kept out a close-range strike from Welbeck.
The Gunners introduced Giroud and it was the Frenchman that had the next clear chance, but the striker could not head a Sanchez cross into the back of the net as it remained 3-2 to the home team with 10 minutes left.
Arsenal pressed for a winner in the latter stages, but some excellent work from Depay wasted precious seconds as Man United held on in the five additional minutes of stoppage time to keep their top-four hopes alive.
There was even time for Van Gaal to send on James Weir as another youngster made his Premier League debut, joining Rashford and Fosu-Mensah, who proved a strong physical presence in the final minutes.
Man United, who will attempt to make it four straight wins in all competitions when they face Watford on Wednesday, are now unbeaten at home to Arsenal in the Premier League for nine matches.