The selection gave the Redskins their second high-profile quarterback of the draft after they took Baylor's Robert Griffin Jr. with the second overall pick.
Draft experts predicted the hard-working Cousins would be a good fit with Griffin and provide beneficial competition in training camp.
The fourth round got under way with a trade Saturday that enabled the Miami Dolphins to grab University of Miami running back Lamar Miller.
The Dolphins sent four picks to the San Francisco 49ers in exchange for their No. 2 selection in Round 4, including three picks in this year's draft and their sixth-round selection in 2013.
Miller was the second-leading rusher in the ACC last season and is considered a good special-teams player; however he only started one season before entering the draft as a sophomore. Some scouts considered him to be too green to adapt to the NFL right away.
Round 4 opened with St. Louis using the 96th overall pick to take Wake Forest wide receiver Chris Givens.