Costa started three games but was forced out of the season opener after one series because of a back injury that plagued him in training camp. He was hurt in the first half against Carolina on Oct. 21.
The Cowboys have used three different starting centers this season with Costa, Ryan Cook and Mackenzy Bernadeau. Cook returned from a knee injury last week against Philadelphia after missing two games.
Costa is the sixth starter the Cowboys have put on injured reserve this season. He will be a restricted free agent after the season.
The Cowboys signed tackle Darrion Weems off Denver's practice squad to take Costa's roster spot. Weems has been with four different teams since coming into the league out of the University of Oregon, signing with Minnesota as an undrafted free agent, then briefly playing for New England, Indianapolis and Denver.
Weems started 22 of 41 games at Oregon, including all 14 as a senior.