The accident happened just after 1:30 a.m. on Lincoln Avenue.
Police say the truck hit the pole and pulled the lines down into the street, knocking out power to nearby homes and businesses.
"It's all high-voltage power lines, fiber-optic cables," Fair Lawn Police Sergeant Sean Macys said. "The fiber-optic cables are going into the bank, and they're all in the ground, so it leads to a lot of damage."
The truck had left the scene by the time police arrived, but authorities said he later turned himself in.
Crews are working to fix the damage, with repairs expected to extend into Thursday.
"It's going to go throughout the night into tomorrow," Macys said. "And the weather isn't helping us."