The video was taken during last night's Europa League quarter-final between the two clubs, which the Gunners won 2-0.
In a statement, Arsenal told Sky Sports News: "We utterly condemn use of this type of racist language and have launched an investigation to identify the culprit.
"We operate a zero tolerance approach, and anyone behaving like this is not welcome at Arsenal and will be banned from matches.
"We have an extremely diverse community of fans who are all part of the Arsenal family and such incidents are rare at Emirates Stadium. We encourage supporters to report any incidents as they happen to stewards or via our matchday alert service."