Drury will make his major league debut at third base and bat seventh in Tuesday's first game against the Colorado Rockies.
Drury, 23, combined to hit .303 with 40 doubles, five home runs, 61 RBIs and 65 runs scored in 130 games with Reno and Double-A Mobile. The 6-foot-1, 215-pounder entered 2015 ranked by Baseball America and MLB.com as Arizona's No. 7 prospect.
Burgos, 24, is 1-2 with two saves in two stints with the Diamondbacks this season. He has a 5.14 ERA in 22 appearances.
Stites, 25, appeared in four games over two stints with Arizona this season and had a 3.86 ERA. He was 1-1 with three saves and a 3.86 ERA in 23 games with Triple-A Reno.
Stites began the season on the disabled list with right elbow inflammation.