Brissett, acquired in a trade with New England earlier this month, looked much more comfortable in his second start, going 17 of 24 with 259 yards and running five times for 14 yards.
Cleveland (0-3) lost its 15th consecutive road game despite being favored for the first time since 2015 and a road favorite for the first time since 2012.
He started the scoring with a 5-yard TD run, broke a 7-7 tie by spinning away from pressure and scooting 7 yards for his second score and then hooked up with T.Y. Hilton on a 61-yard TD pass to make it 21-7 — all in the first half.