Angelillo was part of the Argentine national team that won the 1957 South American Championship. That team, and its forward line in particular, was nicknamed "The Angels with Dirty Faces" (a reference to the then-celebrated "Angels with Dirty Faces" movie) due to both their irreverent style of play as well as to their less than rigorous attitude to training. The nickname followed Angelillo when he, along with fellow national team forwards Omar Sívori and Humberto Maschio, moved from Argentina to play football in Italy after the tournament. There, the trio was also known as "The Trio of Death" due to their clinical ability in scoring goals.