Marcus Rashford and Anthony Martial scored for Manchester United in the first half to give Ole Gunnar Solskjaer's side a shock lead before Nicolas Otamendi ensured a nervy finish for the visitors to the Etihad.
But the game was overshadowed by an ugly incident in the second half when Fred was hit by a missile from the City supporters and one appeared to make a racial gesture in his direction.
A Manchester City statement said: "Manchester City FC are aware of a video circulating on social media which appears to show a supporter making racial gestures during the second half of the match against Manchester United this evening.
"Officials from the club are working with Greater Manchester Police in order to help them identify any individuals concerned and assist with their enquiries.
"The club are also working with GMP regarding an instance of objects being thrown onto the field of play.
"The club operates a zero tolerance policy regarding discrimination of any kind, and anyone found guilty of racial abuse will be banned from the club for life."
‘We keep talking about it every week’ – Solskjaer calls for action over alleged racial abuse
Manchester City found themselves investigating another racism scandal last season when winger Raheem Sterling was subjected to abuse from the crowd referencing the colour of his skin at Stamford Bridge.
The former Liverpool player has also been subjected to racist abuse on international duty and has become a prominent figure on the subject.
And his manager Pep Guardiola was quick to condemn the actions of any supporter found to have been involved in racism.
Guardiola said: "I support the club and I'm going to work to make sure it doesn't happen again."
His opposite number Solskjaer was also disappointed that the game had been overshadowed by yet another incident of abuse.
"We have to stamp it out. We keep talking about it every single week," Solskjaer said.
"He should not be allowed back into a ground, it's as simple as that."