Cloutier began his professional career in 1996–97 with the Binghamton Rangers of the American Hockey League (AHL), New York's minor league affiliate. He assumed the starting position in Binghamton as a rookie, posting a 3.55 GAA and .892 save percentage to be named to the AHL All-Rookie Team. The following season, New York changed their AHL affiliate to the Hartford Wolf Pack; Cloutier was consequently moved to Hartford for the club's inaugural season in 1998–99. By late-December 1998, he earned a call-up to the Rangers as backup goaltender Jason Muzzatti was demoted to the Wolf Pack. At the time of his call-up, Rangers management established that although he would play backup to Mike Richter, they intended on giving him opportunities to play. In anticipation of Cloutier's NHL debut, the team's goaltending consultant, Sam St. Laurent, described his playing style as a combination of both butterfly and stand-up goaltending.