Jones-Drew finished with 136 total yards, establishing a new franchise record with 73 career touchdowns.
The Jaguars (4-9) snapped a three-game losing streak thanks in part to Blaine Gabbert, who threw for 217 yards with two touchdowns and two interceptions.
He hit Marcedes Lewis with a pair of passes good for 77 yards in the win -- the Jags' first under interim head Coach Mel Tucker.
The reeling Buccaneers (4-9), meanwhile, blew a 14-0 lead and committed seven turnovers en route to their seventh straight loss.
Bucs quarterback Josh Freeman re-entered the lineup after missing last week's game, tossing for 181 yards and two interceptions on 16-of-30 passing efficiency.
He also ran for 26 yards and a score.