Dawson's 52-yarder in the second quarter gave the Browns a 3-0 halftime lead and his 53-yard effort early in the fourth -- following an 11-yard, 48-yard drive -- broke a 3-3 deadlock.
Montario Hardesty filled in, gaining 96 yards on 33 carries for Cleveland.
Seattle, meanwhile, was without starting quarterback Tavaris Jackson. and backup Charlie Whitehurst struggled, completing only 12-of-30 passes for 97 yards and an interception.
Ron Burgundy interviews Peyton Manning on SportsCenter
Toddler uninjured after being knocked over by Obama family dog