Gorgen, 23, was 3-2 with a 1.65 ERA and 22 saves in 42 games for Double-A Montgomery this year. The right-hander tied for the Southern League lead in saves and was named to the circuit's midseason and postseason All-Star teams.
Initially drafted by the Rays in the 16th round in 2008, he is 11-4 with a 1.45 ERA in 108 games during his pro career and has held opponents to a .167 average.
Gorgen will be assigned to the D-backs' Triple-A team in Reno.
In 16 games for the Rays since the deal, Qualls has a 6.57 ERA.
Steve Gilbert is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.