Weeden completed 25-of-36 throws, overcoming two first-half interceptions for Cleveland (4-8), which logged its first road victory of the season.
The Browns signal-caller engineered a 14-play, 94-yard drive in the fourth quarter capped by Trent Richardson's 3-yard run with 3:27 left, giving Cleveland a 10-point lead.
Richardson ran for 72 yards on 20 carries as the Browns notched their second straight victory.
Oakland (3-9), meanwhile, fell for a fifth straight game despite a 34-for-54, 351-yard passing performance from Carson Palmer.
He was picked off in the fourth quarter, however, leading to the decisive Cleveland touchdown.
Brandon Myers picked up 130 yards on 14 catches for the Raiders.