Rookie Erik Cernak has impressed the Lightning since filling in for Anton Stralman.
The running back averaged 5.2 yards in the first half, but his second half was remembered by a costly three-yard loss.
The Bucs lost their 12th straight road game started by Jameis Winston, who hasn't led them to a second-half touchdown in consecutive weeks.
The Lightning and Jets went back and forth before finally ending in overtime.
Baltimore committed to wearing out the Bucs out with long sustained drives, mostly through a ground game that piled up 242 yards.
The veteran did his part to stop Lamar Jackson with 12 tackles, two sacks and a fumble recovery.