The American team has gone 4-0 during the first two days of the Olympics.
Ramirez, fighting in the lightweight division, survived a slugfest with Rachid Azzedine of France, winning a decision 21-20.
Ramirez will now meet Fazliddin Gaibnazarov of Uzbekistan, who outpointed Yhyacinthe Abdon of Cameroon 11-6.
In the welterweight division, Spence was a 16-10 winner over Myke Ribeiro of Brazil. Up next for Spence will be India's Krishan Vikas, who had a bye past the first round.
The powerful Cuban squad did not send a fighter into the ring during the first two days of the Olympics.
Two Cubans had bye Saturday and Toledo Lopez also had a bye in the lightweight division Sunday. Cuba does not have an entrant in the welterweight class.