With less than a minute left, the Dolphins had the ball deep in their own end and it seemed the game was heading to a overtime. But Chad Henne's pass attempt was tipped at the line and grabbed by Adams, who returned it to Miami's 2-yard line with 54 seconds left.
Cleveland then ran the clock down and Dawson connected on a 24-yard game-winner for the Browns (5-7), who notched their second straight victory.
Jake Delhomme completed 24-of-34 passes for 217 yards, including a TD toss to Benjamin Watson (100 yards on 10 catches) and Cleveland took advantage of three Henne interceptions in the contest.
Henne ended the day 16-of-32 passing for 174 yards for Miami (6-6).