Van Gaal saw his side take the lead in the 29th minute when Marcus Rashford fired into the top corner from inside the box on his Premier League debut.
And the teenager made it 2-0 moments later when he blasted home from close range.
Arsenal pulled a goal back when Danny Welbeck headed Mesut Ozil’s free-kick into the near post before half-time.
However, Ander Herrera’s deflected shot restored Manchester United’s two-goal lead in the 65th minute.
Ozil pulled another goal back for the Gunners in the 69th minute but the north London side were unable to conjure an equaliser at Old Trafford.
The result left United three points adrift of the top four heading into the final 11 games of the season.
Speaking to BBC Sport after the game, Van Gaal said: “The performance and the result [was pleasing], when you are playing against one of the best Premier League teams and can play with such ambition and performance.
“We played very high in the first half, that is risky against a team like Arsenal but we changed things in the second half and were very compact.”
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