Jussi Jokinen had 2 goals and an assist in his return from an injury, and Erik Cole, Tuomo Ruutu and Eric Staal also scored for the Hurricanes, who couldn't hold a three-goal lead but still won their third straight to improve to 6-0-2 in their last eight.
"I think we know we have things to work on (and) I think everyone knows what they are," said Skinner, an 18-year-old picked No. 7 in the Entry Draft last June and the youngest player active in the League this season. "You just are happy to get out of here with the two points and regroup and come back better the next day."
When you're 14th in the West and desperate to get back into the playoff race, one point isn't enough.
"I'm not happy with the point. We need to win games," coach Brent Sutter said. "It was great that we battled back, and we played like we did in the third period, but we should never have a letdown like we did in the second."
Skinner, who entered with 32 points to lead all rookies, had an early assist before coming through with the only goal in the shootout. He deked Henrik Karlsson before snapping the puck high past him for his team-leading third shootout score of the season.
"I want to be a difference-maker in the game, and that's shooting in the shootout," Skinner said.
Cam Ward stopped a wrist shot from Alex Tanguay and a backhand from Ales Kotalik before Hagman sent a wrist shot into his pads to end it.
Calgary trailed 5-2 late in the second period but reeled off three goals in an 11-minute span. Hagman started the rally by beating Ward with a slap shot at 16:48 of the second period. Glencross cut the deficit to one goal 1:31 into the third, and Bourque tied it at 5-5 by beating Ward with a wrist shot with 12:21 remaining.
"The third period was a good character period by us, but it's so frustrating because when you see what you can do when you stick to the game plan and everyone buys in. You see the results you can have," Calgary center Brendan Morrison said. "But when you don't do that for a full game, it's difficult to win in this league."
Carolina had the only four shots in OT but couldn't cash in on a power play before Karlsson stuffed Jussi Jokinen from point-blank range with about 10 seconds left.
Ward, who along with Staal will play in the All-Star game later this month on home ice, returned after missing two games with a lower body injury and finished with 27 saves for Carolina. Miikka Kiprusoff stopped just four of eight shots for Calgary before he was relieved early in the second by Karlsson, who made 21 saves.
"What (Karlsson) did in net for us was, he battled," Sutter said. "And that's what we want to see with our whole team."
The Hurricanes appeared to have things under control in the second, scoring two goals seconds after power plays expired and building a three-goal lead. Cole and Staal scored early in the period to chase Kiprusoff, and Jussi Jokinen's deflection at 14:50 made it 5-2
Jarome Iginla, named an All-Star for the sixth time earlier in the day, earned the 500th assist of his career on Bouwmeester's goal 4:13 into the game. After Jussi Jokinen tied it, the Calgary captain made it 2-1 at 13:06 when he backhanded rebound behind Ward for his team-leading 18th goal.
Material from team media and wire services was used in this report
1 - 0 CGY
1 - 1 Tie
2 - 1 CGY
2 - 2 Tie
3 - 2 CAR
4 - 2 CAR