Skinner, last season's NHL rookie of the year, has six goals and 12 points in 12 games this season.
Carolina improved to 1-2-1 against Southeast Division rivals. The Lightning opened the season with a 5-1 road victory over the Hurricanes on Oct. 7. Tampa Bay had won three straight against Carolina in Raleigh by a combined score of 15-5.
Carolina's Cam Ward (5-2-2) made 27 saves. Tampa Bay counterpart Dwayne Roloson (2-3-1) stopped 26 shots.
Hurricanes forward Jussi Jokinen sustained an apparent injury to his left knee early in the third period and didn't return.
Carolina outshot the Lightning 19-7 in the first period, but the game was tied 1-1 heading into the second.
Tampa Bay took a 1-0 lead with 4:46 left when Teddy Purcell picked up the puck after Bryan Allen blocked Marc-Andre Bergeron's shot from the high slot.
LaRose tied it with a power-play goal at 18:21. He gathered a loose puck behind Roloson off a rebound of Jay Harrison's shot and tipped it in. LaRose's third career power-play goal extended his scoring streak to three games.
Skinner put Carolina ahead 22 seconds into the middle period, using fancy stickwork in traffic to beat Roloson stick side from the low slot. Joni Pitkanen had the only assist on Skinner's eighth point in eight games against the Lightning.
Carolina killed off a 5-on-3 power play later in the period when Eric Staal and LaRose was whistled for simultaneous penalties.
Skinner pushed the Hurricanes' lead to 3-1 on a rush at 10:38. He took Jokinen's pass from the left circle on the right doorstep and easily beat Roloson.
Vincent Lecavalier nearly brought the Lightning within a goal, but his shot from the left point hit the right post with 3:15 left in the period.
Tampa Bay did make it 3-2 at 14:16 of the third when Brett Connolly picked up a loose puck and poked it past Ward for his first NHL goal.
LaRose then netted an insurance goal at 15:10 on a rush, taking Jiri Tlusty's centering pass and beating Roloson between the pads.
NOTES: Carolina, which opened a three-game homestand, will have 10 of its 15 games in November at home. The last shutout in this series was a 4-0 Lightning road win on Feb. 1, 2007. Bergeron's first-period assist was his 13th point of the season, tying him with Ottawa's Erik Karlsson for the most among NHL defensemen.