Goalkeeper Dibu Martinez raced to one of the rails in front of the Argentina fans to ask police to stop the violence.
Brazil fans, who jeered and applauded Messi before the brawl began, chanted against him when he returned to the pitch.
Argentina leads South American World Cup qualifying with 12 points from five games.
The rivalry between Brazil and Argentina started at the club level and grew after the Argentinians won their first World Cup title in 1978.
The reginal rivals have since played fierce matches in World Cup qualifying and in Copa America, but not in World Cups.
Persons:
— Lionel Messi, Dibu Martinez, jeered, Messi, Diego Maradona, Brazil's
Organizations:
RIO DE, American, Argentina, Police, Inter Miami, South, Argentinians, Copa America, Messi, Brazil's Fluminense, Copa Libertadores, Boca Juniors, ___
Locations:
RIO DE JANEIRO, Argentina, Brazil, Rio de Janeiro, Argentine, Barcelona, Copa, Qatar, Croatia