United were beaten by Villarreal 11-10 in a shootout after a 1-1 draw following extra time in Gdansk.
"At least 70 racial slurs on my social accounts counted so far," Rashford tweeted.
"For those working to make me feel any worse than I already do, good luck trying."
I'm more outraged that one of the abusers that left a mountain of monkey emojis in my DM is a maths teacher with an open profile. He teaches children!! And knows that he can freely racially abuse without consequence...— Marcus Rashford MBE (@MarcusRashford) May 27, 2021
United tweeted after the game players were subjected to "disgraceful racist abuse" and urged users to report any form of abuse or discrimination through their online reporting tool.
A host of players at Premier League clubs have been targeted in the past few months, including United's Anthony Martial, Manchester City's Raheem Sterling, Liverpool's Trent-Alexander Arnold and Sadio Mane, and Chelsea's Reece James.
In February, English football bodies sent an open letter to Facebook and Twitter, urging blocking and swift takedowns of offensive posts, as well as an improved verification process for users.
Instagram has announced new measures and Twitter vowed to continue its efforts after taking action on over 700 cases of abuse related to soccer in Britain in 2019.