With the first deal of the draft, the Browns moved down five spots and acquired Buffalo's No. 9 pick and first- and fourth-round picks next year. Cleveland then swapped picks with Minnesota, acquiring a fifth-rounder before taking Gilbert, who can play on the opposite side of Pro Bowler Joe Haden.
The 6-foot, 202-pound Gilbert had seven interceptions last season, returning two for touchdowns.
Cleveland entered the most important draft in its expansion era with 10 picks, ample ammunition for new general manager Ray Farmer to add talent and depth to a team that went 4-12 last season.