Partizan's Brazilian midfielder Everton Luiz was subjected to monkey chants from Rad fans throughout the match and made an obscene gesture after the final whistle, prompting the home team's players to confront him in the Serbia top tier.
A full scale brawl broke out before staff from both sides eventually managed to separate the players.
Partizan coach Marko Nikolic said Everton was likely to face disciplinary action from the club but urged Serbia's football authorities to punish Rad's supporters.
"Everton should not have reacted but something provoked his outburst, didn't it?", said Nikolic, who was sacked by Slovenian side Olimpija Ljubljana last April for racially abusing one of his own players.
"Both sides in the incident should be punished. These things happen in football but it is now up to the authorities to sanction the offenders," he added after Sunday's match.
While at Olimpija, Nikolic offended the club's Nigerian striker Blessing Eleke by calling him a "black idiot" for his exuberant celebration of an equaliser which meant his team had less time to find a winner in a 1-1 home draw with Zavrc.
Nikolic returned to Partizan in August for his second stint with the club after managing them from 2013 to 2015.
Partizan beat Rad 1-0 thanks to an Uros Djurdjevic goal which kept them second in the 16-team league, six points behind champions and leaders Red Star Belgrade.