He played junior hockey for the Montreal Junior Canadiens in 1971-72 and, after that team folded, joined the Sorel Éperviers for 1972-73 and 1973-74. He was a first all-star in both seasons with Sorel. The Pittsburgh Penguins chose him 27th overall in the 1974 NHL Amateur Draft. Cossette turned pro with the Hershey Bears in 1974-75 and managed to play in 64 games for the Penguins over the next five seasons, scoring 8 goals and 6 assists. He retired in 1980 after a season with the Syracuse Firebirds.