Godfrey was 9-2 with a 3.29 ERA and one save last season with Triple-A Sacramento. He started the year with Oakland and appeared in five games, going 0-4 with a 6.43 ERA. Godfrey, 28, has made eight starts in his 10 major league appearances, all during the past two seasons, and is 1-6 with a 5.09 ERA.
The Red Sox assigned him to Triple-A Pawtucket.
Boston traded Rosario to Oakland on Nov. 28. The Athletics designated him for assignment two days later.