Young went 2-6 with a 6.26 ERA in two stints with the Diamondbacks this season. He made 30 appearances and two starts before being designated for assignment on July 21.
The Indians optioned the 27-year-old, a second-round draft pick from Texas Christian in 2015, to Triple-A Columbus.
In three seasons for Arizona, Young went 11-15 with a 4.73 ERA in 62 outings, including 24 starts.
To make room on the 40-man roster, the Indians transferred right-hander Aaron Civale from the 10-day injured list to the 60-day IL. Civale has been out since June 21 with a sprained finger.
Civale was 10-2 with a 3.32 ERA when he had to be removed from a start against the Chicago Cubs. He's scheduled to pitch a fastballs-only bullpen session on Tuesday, when the Indians get a better sense on when he might be back in their rotation.
Cleveland opens a two-game series with St. Louis on Tuesday.