Blazek, who pitched at three levels a year ago, figures to fill a late-innings role with Nashville.
"We really like his stuff, and he's got a good head on his shoulders," Roenicke said. "He'll be back in the Major Leagues, and he can pitch here for a long time."
The move left the Brewers with 35 players in big league camp.
Chris Gabel is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.