The club issued a statement on Thursday to say they were "disgusted" by the messages received by Martial and Tuanzebe and urged social media companies to take more action to protect users on their platforms.
Captain Harry Maguire posted a message on Twitter to say the squad "will not tolerate" any form of racism while defender Brandon Williams branded the abuse sent to Martial and Tuanzebe as "disgusting behaviour."
The statement issued by United read: "Everyone at Manchester United is disgusted by the racial abuse received by players via social media after last night's game.
"We utterly condemn it and it is encouraging to see other fans condemn this on social media also.
"Manchester United has zero tolerance of any form of racism or discrimination and a long-standing commitment to campaigning against it through our All Red All Equal initiative.
"Identifying these anonymous mindless idiots remains problematic. We urge social media platforms and regulatory authorities to strengthen measures to prevent this kind of behaviour."
Tuanzebe posted a video to his Instagram account later on Thursday with the message: "My RELIGION My COLOUR =FOOTBALL."