Bagent exited in favor of Case Keenum after completing 13 of 22 passes for 169 yards with one TD, one interception and a 79.5 passer rating in the first half.
Watson connected on 11 of 15 passes for 126 yards and a 98.2 rating before being replaced by Shedeur Sanders to start the second half.
The Browns had taken a 10-0 lead on Andre Szmyt's 50-yard field goal midway through the first quarter and receiver Isaiah Bond's 14-yard TD run on a jet sweep on first play of second quarter.
The Bears took a 17-10 lead early in the second half thanks to an impressive effort by running back Salvon Ahmed, who caught a screen pass from Keenum to the left, broke two tackles and zigzagged back across the field for a 39-yard TD.