Juventus made the decision to allow approximately 12,000 children to occupy seats at either end of their stadium, after the club's ultras had been banned for the fixture.
The match would have been played behind closed doors following discriminatory chants towards Napoli supporters during a 3-0 victory on November 10, but Juventus' initiative prevented that scenario.
However, the move backfired with Brkic targeted, and Juve have subsequently been punished by the Italian Football Federation (FIGC).
A statement from Lega Serie A read: "Juventus have been fined €5,000 for their very young supporters having repeatedly insulted a player of a rival team."
Juventus, who lead Serie A, appeared to be heading for a draw against Udinese, but claimed all three points thanks to Fernando Llorente's injury-time winner.
