Juventus fans chanted racist slurs at the Belgian international, prompting him to celebrate and then be sent off.
Roc Nation Sports, Lukaku's representatives, issued a statement urging the Italian authorities to punish Juventus for the racist abuse.
“Tonight's racist remarks made towards Romelu Lukaku by Juventus fans in Turin were beyond despicable and cannot be accepted,” the statement read.
“Romelu deserves an apology from Juventus, and I expect the League to condemn the behaviour of this group of Juventus supporters immediately. The Italian authorities must use this opportunity to tackle racism, rather than punish the victim of the abuse.”
Despite the racist abuse, Juventus coach Massimiliano Allegri downplayed the incident, saying: “It's nothing, just a bit of chaos at the end of a balanced game. We knew it would be difficult. Although they were coming off a difficult period, they are still Inter and proved it tonight.”
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The game saw a total of three red cards, with Juve's Juan Cuadrado and Inter's goalkeeper Samir Handanovic also being dismissed. All three players who were sent off will miss the second leg of the Italian Cup semi-final.