The Browns had to overcome a career-high six TD passes from Carson Palmer, and the game featured more than 1,000 yards of offense.
On the Bengals' final drive, Palmer was intercepted with 20 seconds left on the clock by Browns cornerback Leigh Bodden. Bodden made a diving catch near Cleveland's sideline and managed to stay inbounds at the 27-yard line, sealing the win.
Anderson, starting in place of Charlie Frye, who was traded to Seattle last week, completed 20 of 33 and was intercepted once.
Braylon Edwards caught eight for 146 yards and two touchdowns, and Joe Jurevicius caught two touchdown passes for the Browns (1-1), who finished with 554 total yards and 23 first downs. Kellen Winslow had six receptions for 100 yards and a score.
Palmer went 33 of 50 for 401 yards and was intercepted off twice. Chad Johnson caught 11 balls for 209 yards and two touchdowns for the Bengals (1-1), who gained 531 total yards and 33 first downs.