In 1962 he joined FC Barcelona of Spain, where he was part of the team that won the Copa del Rey in 1963.

In the last years of his career he played for Santiago Morning of Chile and Defensor Sporting of Uruguay where he helped the club to win their first league championship and break the complete dominance of the league by Peñarol and Nacional.

Between 1959 and 1974 Cubilla played 38 games for the Uruguay national team in which he scored 11 goals.

As a coach, Cubilla achieved enormous success with Olimpia Asunción of Paraguay, winning 7 international titles and several national championships.

He also coached Nacional, Peñarol, Defensor Sporting, Danubio (all of Uruguay), Atlético Nacional of Colombia, Newell's Old Boys and River Plate of Argentina and Cerro Porteño and Club Libertad, both from Paraguay.