Nashville took a 2-1 lead early in the second period and could not get another shot past Justin Peters, who finished with 20 saves.
It was Carolina's first win over the Predators in seven meetings.
Skinner's first goal, coming 6:11 into the second, created a 2-2 tie and Riley Nash put the Hurricanes in front with 8:09 left in the middle stanza.
Skinner added a power-play goal in the final minute of the second and tacked on an empty-net goal in the third. He has scored nine times this season.
Marek Mazanec opened in goal for the Predators and gave up four goals on 13 shots over two periods before being replaced by Carter Hutton.