Alvarez defeated Joseph Diaz, Jr., of the United States in one of three losses suffered by the Americans in the boxing ring.
The U.S. contingent numbered nine when the Olympics began and it is now down to four with only the bantamweight class having had two rounds of action.
The victory for Alvarez over Diaz came in the second round of bantamweight boxing and gave the Cuban a spot in the quarterfinals against Brazil's Robenilson Viera de Jesus.
Losses in the two heaviest weight classes in the Olympics further damaged the American hopes for a medal. Both those setbacks came against Russian boxers.
In the heavyweight division, Artur Beterbiev defeated American Michael Hunter and in the super-heavyweight division it was Magomed Omarov over Dominic Breazeale.
The Americans remaining in the boxing tournament include middleweight Terrel Gausha, lightweight Jose Ramirez, welterweight Errol Spence and flyweight Raushee Warren.
Gausha, Ramirez and Spence all won their opening bouts while Warren received a bye to the second round. Warren will enter the ring at the Olympics for the first time on Friday against Nordine Oubaali of France.