The Falcons also agreed to terms with defensive tackle Marlon Davidson, offensive lineman Matt Hennessy, linebacker Mykal Walker, safety Jaylinn Hawkins and punter Sterling Hofrichter.
The rookies are expected to be with the team when the Falcons open training camp on July 28.
Terrell and Davidson, the second-round selection from Auburn, and Walker could win starting jobs in camp.
The Falcons used four of their six draft picks to improve their defense.
Terrell, from Atlanta, is the fourth player the Falcons have selected from Clemson since 2008. He had six interceptions in his career at Clemson.
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