A number of fans were prevented from entering the ground ahead of the Premier League clash and others were ejected during United's 2-0 win, according to Chelsea.
A statement on the club's website read: "At last night's match a large group of Manchester United supporters made unacceptable homophobic chants.
"A number of these away supporters were prevented from entering the stadium and others were ejected during the game.
"This behaviour will not be tolerated at Chelsea Football Club.
"Those supporters will be banned from attending matches at Stamford Bridge in future, and in cooperation with Manchester United we have passed the details of those supporters to them so they can take appropriate action.
"Chelsea Football Club is a diverse and inclusive club and will always take firm action against all forms of discriminatory behaviour."
United said a song sang by some members of the away end ran counter to the club's values.
"Manchester United has a long-standing commitment to equality, diversity and inclusion, embodied through our #AllRedAllEqual initiative," a club spokesman said.
"We were the first club to sign up to the TeamPride coalition and continue to collaborate with Stonewall and other anti-discriminatory organisations in this area.
"Our fans were vocal as always in their support for the team on Monday night and we appreciate that loyal backing. However, this song directed against Chelsea FC – or any other club – by some of our fans runs counter to our values.
"We will continue to engage in campaigns to raise awareness and fight discrimination in all forms."