Johnson ran for 1,652 yards and 10 touchdowns during his junior season with the Hurricanes and cleared more than 900 yards in his freshman and sophomore campaigns. Johnson added 38 receptions for 421 yards and three touchdowns in 2014.
It marks the second straight year Cleveland has selected a running back in the third round. The Browns traded up in last year's third round to take Terrance West.
Johnson is the fourth player selected by the Browns, joining DL Danny Shelton, OL Cameron Erving and LB Nate Orchard.
The Browns have seven picks remaining, including three apiece in the fourth and sixth rounds.