"I feel like sometimes it takes a little time for me to get going and get my confidence," Teravainen said. "I always feel like I can play better."
Gostisbehere made it 4-0 at 15:50, scoring on a one-timer from the left point on a power play.
"They told me my role, they were honest with me from Day One," Gostisbehere said. "That's what I focus on. I'm just happy to get on board this train and hopefully keep it rolling."
Teravainen completed his second NHL hat trick with another power-play goal to push the lead to 5-0 at 11:03 of the third period, burying the rebound of Gostisbehere's shot.
"It's pretty special to score three goals, for sure," Teravainen said. "I lost my grandfather (a week ago), so that's going for him."
Kotkaniemi tipped Brent Burns' shot in the high slot during a power play at 16:34 for the 6-0 final.
Tampa Bay was outshot 23-4 in the first two periods, including being held without a shot in the second.
"No one is happy about what's gone on here in the last little bit," Lightning captain Steven Stamkos said. "When things start to go wrong, it can get in your head a little bit. We've been in this skid for a while now, and we've got to work ourselves out of it as players and coaches. I think people forget we've lost some good players, we've had some young guys come up.
"It's a different group, and you have to figure out the identity of that group. Just because you've had success in years past doesn't automatically translate to, we're just looking to the Stanley Cup Final again because we've been there three straight times. Mental fortitude is part of it, the ability to rely on past experience is part of it. There's so many things that need to come together."
NOTES: Necas has seven points (two goals, five assists) in his past three games. … The Hurricanes have scored seven power-play goals in their past two games after scoring three in a 6-1 win against the Arizona Coyotes on Friday.