The team recalled left-hander Angel Perdomo from Triple-A Nashville in a corresponding move, and also placed infielder Keston Hiura on the COVID-19 injured list.
The Brewers acquired the 38-year-old Axford -- who started his major league career in Milwaukee and had his greatest success there -- from the Toronto Blue Jays for $1 on Monday. The veteran right-hander pitched later that day against the Pittsburgh Pirates but retired just one of the five batters he faced before leaving with an elbow issue.
Brewers manager Craig Counsell said Monday that Axford would undergo an MRI.
Axford began this season as a studio analyst on the Blue Jays' television broadcast crew.
After pitching for Canada in an Olympic qualifier, Axford signed a minor league contract with the Blue Jays on June 24. Axford said his velocity got as high as 98 mph during that Olympic qualifier.
He went 1-0 with an 0.84 ERA in nine relief appearances for the Blue Jays' Triple-A Buffalo affiliate. Axford struck out 14 batters and allowed three walks and two hits in 10 2/3 innings. Opposing batters were hitting just .061 against him.
Axford collected 106 saves during his previous five-year stint in Milwaukee and ranks second to Dan Plesac (133) on the franchise's career saves list.
The Associated Press contributed to this report.