A right-handed starter, Cosart was drafted in 2008 by Philadelphia in the 38th round. The Astros acquired the League City native as one of four minor leaguers they received in the 2011 Hunter Pence trade.

By 2013, he had reached AAA Oklahoma City when the Astros called him up on July 12 for major league action. Cosart shut out the Rays over eight innings in his MLB debut. He went on to post a stingy 1.95 ERA in 10 starts for his rookie campaign, but a low BAbip of .246 on the part of opposing batters hinted that he was having unusually good luck.

That observation was validated the following year when he went 9-7 with a 4.41 ERA. At the July trade deadline, Houston dealt Cosart, along with Austin Wates and Enrique Hernandez, to Miami in return for Francis Martes, Colin Moran, and Jake Marisnick.

Dealt to San Diego in 2016, Cosart's big league career seems to have ended with an 0-5 record there during parts of two seasons. For his career, Cosart compiled a 16-23 record and a 3.98 ERA over five seasons.