A brace withing four minute from Rashford and a third from Ander Herera in the second half was enough to propell United to victory despite goals from Danny Welbeck and Mesut OZil on either side of half-time.
The teenager, who starred for The Red Devils in their 5-1 win over Midtjylland in the UEFA EUropa League in midweek, pounced on a loose ball in the 29th minute before doubling the home side's lead with a header three minutes later.
Ex-United striker Danny Welbeck then pulled on back for The Gunners five minutes before half-time to make for an enteratining close to the first stanza.
Midway through the second half, Rashford's lay-off was curled in by Herrera to restore their two-goal cushion.
Ozil vollied home in the 69th minute, however, Louis van Gaal's side held on for their first win in three Premier League outings, and moved back into fifth place.
The defeat sees Arsenal remain in third position, five points adrift of leaders Leicester City, with Manchester City four points behind them but with a game in hand.