The score was tied 1-1 with 3:31 left in the third period when Tuomo Ruutu got control of the puck behind the net and whipped it to the right circle.
Skinner corralled it there and unleashed a hard wrist shot that beat Carter Hutton through the legs.
He also notched an assist for as the Hurricanes stayed hot in the new year.
Alexander Semin supplied a goal and an assist, and Ruutu finished with two assists.
Anton Khudobin notched his second win in two nights after beating the New York Islanders 3-2 Saturday, turning aside 30-of-31 shots.
Patric Hornqvist scored for the Predators, who dropped a third straight.
Hutton finished with 23 saves.