The Bengals (2-1) blew a 24-7 first-half lead, but two touchdown passes by Dalton in the fourth quarter made the difference.
Griffin's 2-yard run cut the lead to 38-31 with 3:35 remaining, but he threw incomplete on a third-and-50 on the game's last play, and the Redskins fell to 1-2.
Cincinnati's A.J. Green caught nine passes for a career-high 183 yards.
Copyright 2012 by The Associated Press