Kyle Orton also played for the Bears and was good on four of seven throws with an interception.
Cleveland took a 10-0 lead before Grossman hit Brandon Rideau with a 19-yard scoring throw. Robbie Gould's 48-yard field goal as time expired in the first half allowed the Bears to tie it and Gould converted on two more attempts in the second half.
The Browns' Ken Dorsey completed 10-of-20 passes for 92 yards with an interception. Brady Quinn was seven of nine for 65 yards.
Trailing by three points, Cleveland moved to the Chicago two-yard line with 5:32 to play, but Dorsey's fourth-down pass fell incomplete. The Browns regained the ball with 2:34 left and Dorsey was intercepted on the first play of the drive.
Gould's final field goal came with 30 seconds remaining.
Cleveland, which lost all four of its exhibition games, will open the season at home against Dallas. Chicago (1-3) will travel to Indianapolis for its opener.