Prosecutors had demanded a four-year sentence for Ronnie Zape, the 39-year-old duty officer of the Bow Eagle, but the court added an extra year when it found him guilty of failing to assist persons in danger.
He will serve his prison sentence in Norway.
The court noted that it was illegal under Norwegian law for "any individual at the command of a ship or its duty officer to, during a collision or maneuver that has put the life or health of a person in danger, fail to deploy the rescue procedures that are necessary without putting his ship or crew in danger."
Zape was in command last August when the 170-metre (560-foot) Bow Eagle ran down the 26-meter Cistude off France’s Atlantic coast.
Four of the fishermen were killed, and the three others spent more than eight hours in the water before being rescued, after the Bow Eagle steamed away from the scene.
Zape, who faced a maximum sentence of six years in prison, pleaded innocent during his trial last week, arguing that he thought his vessel had had a minor collision with an "unidentified object".
He and a young Filipino sailor, who was also on the bridge at the time of the accident, allegedly initially agreed to keep silent about the collision because, they argued, they thought they had only grazed the French boat.
But the sailor, filled with remorse, defied Zape’s orders to keep mum and told the truth to the second-in-command, who later informed the authorities of the Bow Eagle’s involvement in the accident.
Zape was tried in Norway because the accident occurred in international waters and the Bow Eagle is a Norwegian-flagged ship, owned by the Odfjell shipping group.
According to the conclusions of a French maritime inquiry published in October, the Cistude’s night watch crew was not vigilant enough at the time of the accident.
The trawler, which did not have the right of way over the tanker, did not make any attempt to get out of the tanker’s way, experts said.
However, they stressed the Bow Eagle crew’s "inadequate behavior" in failing to help the fishermen.