NBA executive vice president Joe Dumars announced the suspensions, which also included a one-game ban of Jalen Suggs "for escalating the altercation by aggressively grabbing Rivers around the neck and pulling him to the floor".
Timberwolves forward Jaden McDaniels was fined $20,000 (£16,591) for his role, which included running into the scrum and shoving Bamba. Minnesota's Taurean Prince was also ejected in the incident, but the NBA said Saturday that McDaniels and Suggs "entered the altercation as non-peacemakers".
The wild scenes in Minneapolis erupted in the third quarter with an exchange between Bamba and Rivers in front of the Magic bench.
Orlando centre Bamba left his seat and punches were thrown, sparking a scrum that had security personnel scrambling to separate players.
"Following the incident, Bamba attempted to continue to engage with Rivers in a hostile manner in the corridor outside the locker rooms where he also aggressively shoved a security representative," the NBA said in its statement on Saturday, adding that both players "continued the escalation on social media following the game."
Bamba will begin serving his four-game suspension and Suggs will serve his one-game ban on Sunday when the Magic visit the Charlotte Hornets.
Rivers will begin serving his suspension on Sunday when Minnesota host the Denver Nuggets.
