Jones-Drew rushed for 132 yards on 24 carries and David Garrard shook off three interceptions to gain 254 passing yards, hurling a pair of TD passes in the victory.
The Jaguars (6-4), who reeled off their third straight win, also got five receptions for 66 yards and a touchdown from Marcedes Lewis.
Cleveland (3-7) dropped its second straight game. Colt McCoy passed for 241 yards and a touchdown, but he committed a key turnover with 13 seconds left when he picked off.
Peyton Hillis hauled down six passes for 95 yards and a score and also rushed for 48 yards on 21 carries for the Browns.
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