Burnett, 25, pitched three years in relief for the Minnesota Twin before moving on to Baltimore. He was designated for assignment last Thursday after appearing in two games for the Orioles, giving up four hits and three runs. He's 8-11 with a 4.73 ERA in his career.
Burnett will replace Kyuji Fujikawa on the roster after he was put on the disabled list for the second time this season. Fujikawa aggravated a forearm strain in the Cubs' 5-4 victory on Sunday over the Cincinnati Reds.
Fujikawa's previous stint on the DL was due to the forearm strain.
"We're cautiously optimistic that's it's going to be the same thing he had before," GM Jed Hoyer said. "That's certainly our hope going into the MRI. We'll know more tomorrow."
Hoyer said this wasn't the first time he considered Burnett.
"Alex had a solid season last year in Minnesota," Hoyer said. "A guy we liked and talked a couple of other times about acquiring. Given Fujikawa's injury, we felt it was the right time to do it."