It was approaching 2am at PNC Arena in Raleigh, North Carolina, when Tkachuk at last brought proceedings to a close, giving the Panthers a 1-0 lead in the best-of-seven series.
Sam Bennet picked up the puck along the wall and passed to Tkachuk, whose wrist shot from the right face-off circle got past Carolina goalie Frederik Andersen.
"Probably my favourite one so far in my life," Tkachuk said of the goal.
"Big to not let it go to five overtimes," he added with a relieved chuckle.
It was the longest game in the history of both teams and the 15th four-overtime game in NHL history.
Aleksander Barkov and Carter Verhaeghe each had a goal and an assist, Anthony Duclair had two assists, and Sergei Bobrovsky made 63 saves for the Panthers, who notched their seventh straight road win of the Stanley Cup play-offs.
"Bob (Bobrovsky) played great, everybody followed - total team effort for two games, basically," Tkachuk said.
Seth Jarvis had a goal and an assist, Stefan Noesen scored, and Andersen made 57 saves for the Hurricanes.
It looked like the Panthers had sealed victory at 2:35 of the first overtime when Ryan Lomberg - who missed the previous eight games with an injury - stripped the puck from Jalen Chatfield, turned and scored.
But a video review determined center Colin White had a "significant presence in the crease" and made contact with Andersen, impairing his ability to play his position.
Jarvis then came close to scoring the game-winner at 12:31 on a power play but his put-back shot went off the crossbar.
Carolina had taken a 1-0 lead on a power-play goal with 12 seconds left in the first period. Sebastian Aho held the puck near the right post before passing to Jarvis for a one-timer from the high slot.
All-Star center Barkov tied it 1-1 at 15:28 of the second, scoring from the left circle after a pass from Duclair, and Verhaeghe put Florida up 2-1 with a wrist shot off the left post at 17:43.
The Hurricanes knotted the score at 2-2 on Noesen's power-play goal at 3:47 of the third period.
Florida, the last team to book their Stanley Cup play-offs berth, are in the conference final for the first time since 1996.
The Hurricanes, making their first appearance in the conference final since 2019, host Game Two on Saturday.
The winner of the series will play either Dallas Stars or Vegas Golden Knights in the Stanley Cup Final.
