Arsenal wasted a big chance to turn over the depleted hosts and the London team stayed third in the table, five points behind leaders Leicester City as United climbed to fifth.
The 18-year-old Rashford struck twice in three first-half minutes before former United striker Danny Welbeck pulled one goal back for Arsenal. Rashford then set up Ander Herrera for United’s third before Mesut Ozil reduced the deficit.
With a dozen senior players unavailable through injury, including captain Wayne Rooney, the under-pressure Van Gaal gave a first league appearance to Rashford, who responded brilliantly at Old Trafford.
After 29 minutes he was in the right place to capitalise as Arsenal defender Gabriel could only half clear Guillermo Varela’s right-wing cross.
Three minutes later he headed United’s second goal after Jesse Lingard was given too much space to cross from the same area.
Arsenal, needing a win to keep pace with local rivals Tottenham in pursuit of Leicester, were given hope before halftime when Welbeck — who once played for the same Manchester boys club that produced Rashford — nodded in Ozil’s free kick.
It was the German’s 18th assist of the season and he added a goal of his own in the second half after Rashford had held up the ball intelligently before passing to Herrera whose 20-metre shot was deflected in off Arsenal defender Laurent Koscielny.
Van Gaal — who delighted the home crowd by throwing himself to the ground to illustrate a foul on Herrera — sent on two more young debutants in defender Tim Fosu-Mensah and midfielder James Weir.
United’s third win in seven days after beating Shrewsbury Town 3-0 in the FA Cup and Denmark’s Midtjylland 5-1 in the Europa League had the Old Trafford fans roaring their approval in a difficult season.
“It’s unbelievable,” Van Gaal told Sky Sports about Rashford’s four goals in four days.
“I can understand it last Thursday against Midtjylland when you come in without any pressure but the second time is always more difficult and he did all the things I asked of him.
“Four goals in two matches is an average that’s unusual.”
Arsenal manager Arsene Wenger tried to remain upbeat after another disappointing result that cast doubts over his team’s ability to win a first league title since 2004.
“I cannot fault the commitment and desire but unfortunately we conceded two easy goals,” Wenger said.