Rosenbaum attended Loveland High School in Loveland, Ohio, and Xavier University, where he played college baseball for the Xavier Musketeers baseball team. The Washington Nationals selected Rosenbaum in the 22nd round of the 2009 MLB Draft. The Colorado Rockies selected Rosenbaum from the Nationals in the 2012 Rule 5 draft. The Rockies returned Rosenbaum to the Nationals before Opening Day in 2013.
Rosenbaum underwent Tommy John surgery to repair a torn ulnar collateral ligament of the elbow in May 2014. He was cleared to resume throwing in November. During the 2014-15 offseason, the Nationals traded Rosenbaum to the Boston Red Sox for Dan Butler.
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