The change was announced after Moore got the win Thursday as the Rays beat Minnesota.
Moore is 13-3 and tied with Detroit's Max Scherzer for the league lead in victories. The 24-year-old Moore has a 3.44 ERA in 19 starts.
Darvish was put on the disabled list Wednesday because of a mild strain in his upper back. He is 8-4 and leads the majors with 157 strikeouts.
Darvish is a two-time All-Star. The right-hander plans to attend the All-Star festivities in New York. The All-Star game is Tuesday night at Citi Field.