Bedrosian was traded by the Braves to the Phillies in the off-season and was converted to a reliever before the 1986 season. In his first year in relief, he saved 29 games. His best season came in 1987 when he posted a 5–3 record for the Phillies with a 2.83 earned run average, recorded a league-leading 40 saves, and was named the National League Cy Young Award winner. Since Bedrosian, only three other relievers (Mark Davis, Dennis Eckersley and Éric Gagné) have won Cy Young honors.