Elvis Merzlikins made 28 saves for Columbus (30-17-10), which is 9-1-2 in its past 12 games. The Blue Jackets, who were without defenseman Seth Jones (ankle) and forward Cam Atkinson (lower body), hold the first Wild Card into the Stanley Cup Playoffs from the Eastern Conference.
"It felt like a playoff game a little bit," Columbus defenseman David Savard said. "Those types of games are fun to be a part of."
Kucherov, who has 20 points (10 goals, 10 assists) during an 11-game point streak, won it when he one-timed a pass from Victor Hedman. Brayden Point pressured defenseman Zach Werenski behind the Blue Jackets net, stole the puck and passed to Hedman in the left circle.
"I [screwed] that one up. That's about it pretty much," Werenski said.
Kucherov gave the Lightning a 1-0 lead at 2:29 of the first period with a redirection of Mikhail Sergachev's shot during a 4-on-4.
Oliver Bjorkstrand tied it 1-1 at 3:44 of the second period. He got a partial break off a pass from Pierre-Luc Dubois and scored with a backhand shot for his 18th goal.
The Blue Jackets outshot the Lightning 15-6 in the second.
"We had to regroup, hold onto the puck a little more and make smarter plays," Bjorkstrand said. "I thought we did a good job of that in the second and third periods."
Columbus forward Liam Foudy nearly broke the tie in his NHL debut at 11:18 of the second period, but his shot from the left circle was saved by McElhinney's right pad.
"I was hoping and praying it was in," Foudy said.
He said he was able to adjust to the pace after a few shifts.
"Then you realize it's just another hockey game, don't overthink it too much and just go out there and play your game," he said.
The Blue Jackets again almost took the lead with 5:29 remaining in the third period, but forward Boone Jenner hit the crossbar.
Merzlikins forced overtime when he stretched out and made a glove save on Kucherov's one-timer with 1:18 left.
"It was two pretty good hockey teams that went at it," Lightning coach Jon Cooper said. "It's probably fitting that it went to overtime."