Price took the roster spot of right-hander Joe Kelly, who was removed due to a right biceps strain.
Kelly was a critical middle-innings piece for the Dodgers, key in bridging the gap to the back-end trio of Brusdar Graterol, Blake Treinen and Kenley Jansen.
The Dodgers also finished the postseason without first baseman Max Muncy (left elbow dislocation) and third baseman Justin Turner (left hamstring strain).
"We were getting burned with injuries," said Dodgers outfielder AJ Pollock after the Game 5 victory. "It's tough seeing your guys go down to injuries, guys that are putting their heart and soul into it, and then to have to come out of a game, it's tough. It's tough on them. It's tough on the teammates.
"But [our] bullpen stepped up. And ... it was a good job of the offense, pitching, defense to regroup and play that kind of baseball."