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 LeagueCy Young Award winner. Since Bedrosian, only three other relievers, Mark Davis, Dennis Eckersley and Éric Gagné, have won Cy Young honors (Davis won the National League award in 1989 while with the San Diego Padres; Eckersley won the American League award and was the American League's Most Valuable Player in 1992 with the Oakland Athletics; Gagné won the National League award in 2003 with the Los Angeles Dodgers).
He was traded to the Giants during the 1989 season to help their pennant drive that year. In 1990, he won the Willie Mac Award, voted upon by his teammates, honoring his spirit and leadership.
Currently, Bedrosian resides in Newnan, Georgia, where he served on the Coweta county board of education, through 2010, and is an assistant baseball coach at East Coweta high school. In 2008, Bedrosian was inducted into the Coweta Sports Hall of Fame.
Bedrosian has two sons: Cameron, who played for East Coweta High School and was drafted by the Los Angeles Angels in the 2010 MLB Draft; and Kyle, who played for Mercer University and is a pitching coach at local baseball academy Home Plate.