Rashford found the net twice in the space of three minutes, before former Red Danny Welbeck headed home to put the visitors back into the game.
United restored the two-goal advantage after Rashford picked out Ander Herrera, whose slot deflected in and Mesut Ozil's volley gave the Gunners hope, but United held on.
Another steady display at the back
David De Gea (7 out of 10) - The Spanish international made his return to the starting XI after a couple of games out. He was off his line quickly to deny Monreal in the opening exchanges, but was left helpless for both goals.
Guillermo Varela (7 out of 10) - The youngster is certainly making the most of his chance in the first team. He struggled early on against Sanchez, however, he played his part in both first-half goals from Rashford after getting forward down the flank.
Michael Carrick (6 out of 10) - Maintained his place at the back as injuries continue to plague the side, he struggled initially with the pace of the opposition and was forced to make a challenge on Welbeck during the first-half.
Daley Blind (8 out of 10) - A bit of an unsung hero in the last few games, Blind doesn't really have any real pace about him, but his positional game was superb, as always, and he stopped Arsenal playing their normal game.
Marcos Rojo (6 out of 10) - Rojo started for the first time in 2016, he made an appearance against FC Midtjylland as a substitute. He had an early scare, but kept Walcott quiet, which seen the winger subbed off at half-time. Rojo was replaced by Fosu-Mensah after 55 minutes.
Midfield partnership starting to take shape
Morgan Schneiderlin (6 out of 10) - Was at fault for Welbeck's goal which allowed Arsenal back into the game. He also gave the ball away in the build-up to their second, but his partnership with Herrera seems to be blossoming.
Ander Herrera (7 out of 10) - With Schneiderlin happy to sit and defend in front of the back line, it gives Herrera the freedom to go forward. He stuck to his tasks well and managed to get the third in the game albeit off Koscielny's chest.
Front four lively going forward again
Juan Mata (7 out of 10) - The Spaniard caused Arsenal numerous amounts of problems in the space between their midfield and back line. His play was clever and he also played his part in the decisive third goal.
Jesse Lingard (7 out of 10) - Dangerous as ever down the flank, Lingard played a pivotal role in both of Rashford's goals, including a delightful cross for his second. Never fails to put in a good performance and is a stern favourite with van Gaal now.
Memphis Depay (7 out of 10) - Memphis, for the second game running, looked lively on the ball and a threat going forward. His final ball may well have been better, but the tricks and flicks show that his confidence is starting to come back.
Marcus Rashford (9 out of 10) - It is hard to imagine a better four days than Rashford has just had. He scored two goals on his European debut and two more and an assist on his Premier League debut against Arsenal, he is definitely one to look out for.
Timothy Fosu-Mensah (6 out of 10) - Came on in place of Rojo at the back and it was a promising debut for the youngster.
Adnan Januzaj (6 out of 10) - Replaced Rashford late on, and he looked lively during his spell in the final 10 minutes of the game.
James Weir (4 out of 10) - Was brought on to wind the clock down, replaced Herrera with seconds to play.