The first-round bout was not close with Yeleussinov posting a 19-9 decision. He will meet Mehdi Toloutibandpi of Iran in the second round on Saturday.
American boxers had won their first four bouts before light-heavyweight Marcus Browne lost his opening-round appearance on Monday. Seven U.S. fighters remain in the competition, three of whom have yet to enter the ring.
Raushee Warren received a first-round bye in the flyweight class and neither heavyweight Michael Hunter nor super-heavyweight Dominic Breazeale has seen action.
The traditionally powerful Cuban team is made up of eight boxers and both of those who had bouts Tuesday had no trouble advancing.
Light-flyweight Yosbany Veitia Soto was a 17-7 winner over Billy Ward of Britain and will now take on Zou Shiming of China in the second round.
Cuba's light-welterweight, Roniel Iglesias Sotolongo, scored a 20-9 decision over Cesar Villarraga of Colombia. The Cuban's second-round opponent will be Brazilian Everton dos Santos Lopes, who received a first-round bye.
All eight boxers from Cuba are still alive in the Olympic tournament.