The incident happened before half-time, after Chilean fans began to insult the Barcelona attacker’s family.
Messi’s 28-year-old brother Rodrigo, was hit by one fan after a feisty exchange. The entire family were eventually moved to a television cabin at half-time.
The family of another Argentine star, Sergio Aguero, were also assaulted by another set of La Roja supporters.
Chile eventually beat their South American rivals 4-1, after 120 minutes of goalless football.