Kopech, the top pitching prospect in the organization, is expected to be ready for spring training in 2020. The surgery was performed by Dr. Neal ElAttrache in Los Angeles.
Kopech, 22, was 1-1 with a 5.02 ERA in four starts this season. The right-hander was placed on the 60-day disabled list with a torn elbow ligament Sept 8.
"Michael understands intellectually what it's going to require to come back," manager Rick Renteria said. "We're confident that he'll be able to do that and become the kind of pitcher we know he can be."
Kopech was acquired in the trade that sent star pitcher Chris Sale to Boston following the 2016 season. He was drafted by Boston in the first round in 2014.