Chamberlain signed with the Indians on Dec. 1 as a free agent.
"He accepted the challenge," Francona said Saturday. "He came out in January and been a teammate since day one. He has really thrown the ball well this spring. He likes to compete and wants to win. Joba will be a real valuable piece."
In eight innings this spring, the right hander has given up seven hits and three runs. He walked five and struck out seven and has an ERA of 3.38.
Chamberlain spent last season with Detroit, Toronto and Kansas City. The 30-year-old has a 25-21 career record with a 3.87 ERA. He has seven saves in 365 appearances over nine seasons.
Cleveland informed left-handed pitcher Tom Gorzelanny that he wouldn't make the team. Gorzelanny signed a minor league contract in December.