Fiset was drafted in the 2nd round, 24th overall, in the 1988 NHL Entry Draft by the Quebec Nordiques. He played 34 minutes in 6 games in his rookie season with Quebec, playing behind Ron Tugnutt, as well as a host of other goalies who got time that season. Fiset got his first NHL victory on October 29, 1991 against the Winnipeg Jets in Le Colisee de Quebec, by a score of 7–2. Over the next few years, Fiset bounced from the NHL to the Nordiques' minor league affiliates, slowly gaining more and more time in net for the Nordiques. After the 1992–93 season, Ron Hextall was traded to the New York Islanders, opening the door for Fiset to be the starter. Fiset did not disappoint. Within two years, Quebec finished first in the Eastern Conference, during the shortened 1994–95 lockout season.