Fitzpatrick completed 22-of-35 passes for 208 yards for the Bills (2-1), who broke an eight-game road losing streak.
Seven of his completions went to Johnson for 61 yards. His 9-yard scoring grab with 9:08 left in the game gave Buffalo its final 10-point cushion.
T.J. Graham and C.J. Spiller also caught touchdown passes, but Spiller left the game with a shoulder injury. He is likely to miss next week's contest against New England, Coach Chan Gailey said after the game.
Rookie quarterback Brandon Weeden was picked off twice in the fourth quarter for the Browns (0-3) while Trent Richardson was held to 27 yards rushing and a score on 12 carries.