Sesúmaga was born in the Basque Country and started his career with Arenas de Guecho in 1914. After losing the Copa del Rey (Spanish Cup) final to Madrid FC in 1917, he was on the winning team in 1919, when he scored thrice against FC Barcelona in 1919. This induced Barcelona to sign him for the next season. The forward won the Copa del Rey final in 1920 to become the second player to win this title in two successive years for different clubs (after René Petit). He left FC Barcelona after only two seasons to join the Asturian club Racing de Sama as playing coach, but as the team was not allowed to participate in offical competitions he moved to Bilbao to join Athletic Club. Here he won his third Copa del Rey in 1923.
At the 1920 Olympics, where he was in the line-up of the first-ever match played by the Spanish national team, he was a member of four of the five matches and scored twice against Italy as well as against the Netherlands. During his stint at Racing de Sama he played thrice for the national team, making him the only international for this club. In total Sesúmaga earned eight caps, but the four goals at the Olympics were his only international goals. He had to quit football in 1924 due to illness and died in 1925 aged only 26. His younger brothers Críspulo and Fídel both also played for Arenas de Guecho in Copa del Rey finals and also in the 1929 established La Liga (Spanish League).
Spain