Right-hander Matt Stites was optioned to Triple-A Reno and infielder Phil Gosselin, who has a fractured left thumb, was moved to the 60-day disabled list.
Godley, 25, was acquired by the D-backs in December in the trade that sent catcher Miguel Montero to the Cubs. Godley spent the first two seasons of his professional career as a reliever, but the D-backs have worked on converting him into a starter this season.
The right-hander was 8-3 with a 2.27 ERA in 14 games for Class A Advanced Visalia this year. He then made three starts for Mobile, going 1-1 with a 5.14 ERA.
While there were several candidates to make Thursday's start in the Minors, as well as long reliever Josh Collmenter, D-backs manager Chip Hale said the organization felt Godley was the best choice.
"We felt like this guy has kind of epitomized what we're trying to get through as an organization," Hale said. "He's a big competitor and the way he pitches is kind of the way we're looking for."
Anderson is likely to miss several starts, and Hale said Godley will likely have that spot in the rotation until Anderson returns. Godley hasn't pitched since July 11, when he gave up two runs on six hits in 4 2/3 innings for Mobile.
Godley was a 10th-round pick in the 2013 Draft by the Cubs after pitching for the University of Tennessee.