Buffalo's Rian Lindell was wide right on a 45-yard attempt in the final minute, sealing the Browns' victory.
Dawson made all five field goals he attempted for Cleveland (4-6). Braylon Edwards had eight receptions for 104 yards, and Brady Quinn completed 14-of-36 passes for 185 yards to earn his first career win in his second NFL start.
Marshawn Lynch led the Bills offense, rushing for 119 yards on 23 carries and catching 10 passes for 58 yards and a touchdown.
Trent Edwards threw three interceptions in the first quarter for the Bills (5-5), who lost their fourth straight to fall back to .500 after a 5-1 start.