Born in Mexico City, Carbajal became a professional footballer with the local Club España in 1948, after having been in the squad that participated at the Olympic tournament in 1948. After the disappearance of España in 1950, he joined Club León, where he would remain until the end of his career.
One of the greatest players of all time to come from Mexico, Carbajal made his international debut in the Maracana Stadium in Rio de Janeiro on June 24, 1950 against World Cup hosts Brazil. He was the youngest goalkeeper to play in that tournament, where he saw action in Mexico's three matches. He appeared in one match at the 1954 World Cup and in three at the 1958 tournament; at the 1962 World Cup in Chile, he became the first footballer ever to appear in four World Cups, also helping his team win its first ever World Cup match. When they defeated Czechoslovakia 3-1 in the first round. Four years later Carbajal established another mark with his fifth World Cup appearance. That record was equaled by German player Lothar Matthäus in 1998 and by Italian goaler Gianluigi Buffon in 2014. In total, Carbajal appeared in 48 international matches for Mexico. In 11 World Cup matches from 1950 to 1966, he conceded 25 goals, a record that was tied by Saudi goalkeeper Mohamed Al-Deayea in 2002.