[4][5][6] The New York Yankees selected Cotham in the fifth round of the 2009 Major League Baseball draft.

[9][10] Cotham returned to the New York-Penn League in 2011, allowing four runs on 21 hits and nine walks in 21 innings.

[9] He pitched in the Arizona Fall League that year, allowing nine runs but striking out 14 in 13.1 innings out of the bullpen.

[19] On the same day, he made his Major League debut with the Yankees, pitching one and two-thirds innings, giving up two hits, and striking out four.

[23] Cotham appeared in 12 games for the Yankees, allowing 14 hits and eight runs in 9.2 innings of work.

On December 28, 2015, the Yankees traded Cotham, Eric Jagielo, Rookie Davis, and Tony Renda to the Cincinnati Reds, in exchange for Aroldis Chapman.

Cotham underwent season-ending knee surgery in August,[27][28] finishing the 2016 season with an 0–3 win–loss record and a 7.40 ERA.