In 1979-80, Sauve led the NHL in Goals Against Average (2.04), and along with Edwards, was the co-winner of the Vezina Trophy. Sauve was fifth in GAA the year after but was traded during the 1981-82 campaign to Detroit. His stint with the Red Wings was a short one, as he resigned with Buffalo as a free agent after the season ended.
Sauve played three more years with the Sabres, winning the William M. Jennings Trophy with Tom Barrasso in 1984-85, which was the last year he was with Buffalo. Sauve was traded to Chicago, and he left the Sabres with 119 Wins with a 3.21 GAA.