Striker Zian Flemming scored a 93rd minute header to earn a precious point for Scott Parker’s side at Stamford Bridge.
After the game, however, Burnley midfielder Mejbri, 23, revealed he had been racially abused on social media.
Burnley have since released a statement condemning the abuse and firmly backing their young midfielder.
The statement says: “Everyone at Burnley FC is disgusted by the online racist abuse directed at Hannibal following today’s Premier League fixture.
“There is no place for this in our society and we condemn it unreservedly.
“The Club continues to be unequivocal in its stance - we have a zero-tolerance approach to any form of discrimination.
“The Club has reported the post to Instagram's parent company, Meta, and expects strong support from them, together with the Premier League and the Police, and will work to ensure that the individual responsible is identified and investigated.
“Hannibal will receive the full backing from the Club and from the Burnley fans who we have already seen condemning the abuse.
“There is no room for racism.”