It was Kucherov's sixth goal of the postseason.
Defenseman Mikhail Sergachev was credited with the lone assist on the sequence.
Kucherov added another goal with 8:35 left in regulation.
Point won an offensive zone draw that allowed the Russian star to beat Price with a shot upstairs.
For his part, Point collected his second helper of the game on the play.
Forward Steven Stamkos scored Tampa Bay's fifth goal with 1:10 left in regulation on the power play.
The one-timer from the Lightning captain beat Price under the arm.
Kucherov and Point collected the assists.
That's the way this one finished up.
Price made 21 saves in the loss.
Game 2 will be played on Wednesday night at AMALIE Arena. Puck drop is scheduled for 8:00 p.m. ET.