Point gave the Lightning a 1-0 lead at 14:24 of the first period, scoring off a backdoor pass from Ondrej Palat.
Teuvo Teravainen tied it 1-1 at 4:30 of the second period with a one-timer from the hash marks off a pass from Jordan Staal.
Jesper Fast gave Carolina a 2-1 lead at 5:09 with a sweeping shot from the crease.
"We wanted to come out harder than we did in the first," Fast said. "We played the right way, got a goal. We're up 4-2, had the momentum, and it went pretty quick there when they scored three straight goals. It's tough, but we have to find a way to bounce back. Unfortunately, we didn't."
Stamkos tied it 2-2 with a power-play goal at 9:54.
Dougie Hamilton scored from the point at 10:35 to put the Hurricanes ahead 3-2, and Slavin scored at 12:41 to make it 4-2.
"We score four on [Vasilevskiy], we need to win that game," Brind'Amour said. "He's probably not going to let four in again. I just think, again, the other day, we weren't sharp, then we got going. We had some good momentum and the penalties killed us."
Kucherov made it 4-3 with a power-play goal at 14:38. Johnson tied it 4-4 with a wrist shot from the top of the right face-off circle at 17:10.
Stamkos scored on a one-timer off a pass from Kucherov at 19:37 for the Lightning's third power-play goal of the period to make it 5-4.
"Eight goals scored, probably the craziest playoff period you are going to see the rest of the way," Stamkos said. "Certainly exciting for the fans, but I give our guys credit, we go down two goals on a couple of unfortunate bounces, I didn't think we were really playing our game. We just kind of talked about staying in the moment, we get a couple of power-play goals late and capitalized."
Kucherov extended the lead to 6-4 at 6:01 of the third period with a one-timer off a pass from Palat.
"He was just making plays all over the place," Lightning coach Jon Cooper said of Kucherov. "I thought he might've been the best player on the ice tonight. When 'Kuch' is doing some of the stuff he did tonight, he's borderline unstoppable. It's great to see him do what he did tonight because we definitely needed him."