The Raiders (2-4) erased a 10-point fourth-quarter deficit to force extra time.
The Jaguars had the ball first in overtime, but coughed it up when Cecil Shorts fumbled after a third-down catch at the Jacksonville 21-yard line.
One play later, Janikowski connected on his fourth field goal of the contest, allowing Oakland to snap its two-game losing streak.
Oakland's Carson Palmer completed 26-of-46 passes for 298 yards and a touchdown, and ran for another score.
Jacksonville (1-5), meanwhile, had to play much of the game without star running back Maurice Jones-Drew (left foot) and starting quarterback Blaine Gabbert (left shoulder), both of whom left with injuries.
Chad Henne came in and was limited to 9-for-20 passing for 73 yards.