Born in Clackamas, Oregon, Rosscup attended Forest Grove High School. Rosscup later attended Chemeketa Community College for two seasons, before being drafted in the MLB. During his freshman season at Chemeketa, Rosscup held a 1.55 ERA and earned numerous honors. 
Tampa Bay Rays
Rosscup was drafted by the Tampa Bay Rays in the 28th round of the 2009 Major League Baseball Draft out of Chemeketa Community College. The Rays traded Rosscup with Matt Garza and Fernando Perez to the Chicago Cubs for Sam Fuld, Chris Archer, Hak-Ju Lee, Brandon Guyer, and Robinson Chirinos in 2011. 
Rosscup was called up to the majors for the first time on September 3, 2013. In ten games he had an ERA of 1.35, but started the 2014 season at Triple A Iowa Cubs.  Rosscup was called up again on April 16, 2014, to be available for the Chicago Cubs day-night doubleheader. 
On September 23, 2014, Rosscup was credited with his first career win in a walk off Cubs victory against the St. Louis Cardinals.