Goswami was born on January 15, 1938, in Kishoregunj, Bengal. During the golden era of Indian football, Chuni Goswami was one of the mainstays of the national team. Goswami started his professional career with the Mohun Bagan Club in 1954. In the 1950s and 60s, India did well in the international stage riding on the skills of Goswami and a host of capable players. The highpoint of Chuni Goswami`s career was the football gold at the 1962 Asian Games in Jakarta. In the final, the Goswami-led team defeated South Korea 2-1, with goals coming from P.K. Banerjee and Jarnail Singh.

Goswami captained the Mohun Bagan team from 1960 to 1964. During his stint, the club did exceedingly well in various national competitions including the prestigious Durand Cup. In honor of his services to the game, Chuni Goswami was bestowed with the Arjuna Award in 1963 and the Padmashri in 1983.