In exchange, the Browns, who also parted with a seventh-round pick (No. 229), will receive the Texans' second-round pick (No. 51), fourth-round pick (No. 116) and sixth-round pick (No. 195).
The Texans selected Mississippi State linebacker Benardrick McKinney with the 43rd pick.
Cleveland holds nine more picks in the 2015 NFL Draft after making two in Thursday's first round. The Browns have three apiece in the fourth and sixth rounds, which are set for Saturday.