To make room for Putnam on the 40-man roster, the Cubs placed reliever Kyuji Fujikawa on the 60-day disabled list.
Putnam, 25, is 1-1 with four saves and a 3.26 ERA in 17 relief appearances with Iowa this season. Putnam was one of the last cuts in spring training and has made 10 career relief appearances in the major leagues.
"I had a pretty good spring and obviously was disappointed I wasn't able to break camp with the team," Putnam said Thursday morning before the Cubs played the Chicago White Sox. "It's one of those things you have to keep on keeping on."
Burnett was sent down after being picked up off waivers from the Baltimore Orioles earlier in the week. He pitched a scoreless inning against the Sox in his Cubs debut on Wednesday.
Fujikawa is scheduled to undergo Tommy John surgery after injuring himself in Sunday's win in Cincinnati.