After an end-to-end 3-on-3 extra period, Kucherov finished it on a breakaway against Canucks goalie Jacob Markstrom for his 16th of the season.
Alex Killorn and Valtteri Filppula also scored for Tampa Bay, and Andrei Vasilevskiy stopped 21 shots.
Bo Horvat had a goal and an assist, and Sven Baertschi also scored for Vancouver. Markstrom finished with 32 saves in his seventh straight start.
Tied 1-1 through 40 minutes, the Lightning took the lead with 7:40 left in the third just as a Vancouver penalty expired. Filppula's shot deflected off the foot of Canucks defenseman Alexander Edler and past Markstrom for his fifth of the season.
The Canucks challenged the play for goaltender interference, but the play stood up after video review.
Horvat tied the score with 2:50 left in regulation with his sixth of the season on a shot off the rush that sneaked past Vasilevskiy, who got the start as the Lightning gave No. 1 goalie Ben Bishop the night off.
Tampa Bay came from two goals down in the third period to beat Edmonton 3-2 a night earlier, and grabbed a 1-0 lead 3:41 into this game when Killorn finished off a nice give-and-go with Kucherov by outwaiting Markstrom for his seventh of the season.
The Canucks, who beat the Lightning in December despite being short-handed 10 times, tied the score on the power play with 4:36 left in the first on a highlight-reel effort. Horvat sped into the Tampa Bay zone and made a slick deke on Lightning star defenseman Victor Hedman before sliding a pass to Baertschi, who tapped in his seventh.
Vancouver has been searching for secondary scoring behind Henrik and Daniel Sedin for much of the season, and Baertschi now has five goals and two assists over his last seven games, while Horvat has four goals and an assist over his last three.
The Lightning had a great opportunity to snap the deadlock in the second when Edler was assessed a double minor for high sticking, but Tampa Bay never really threatened Markstrom before another power play later in the period also fizzled.
Vancouver rookie forward Jake Virtanen had a spirited game in his return to the lineup following the world junior hockey championship. The 19-year-old took some heat for his undisciplined play in Canada's quarterfinal loss to Finland, but was the Canucks' best player early and drew the penalty that led to Baertschi's goal in the first. ... Ryan Miller dressed as Vancouver's backup after missing the last six games with a groin injury. ... Canucks forward Jannik Hansen sat out with an upper-body injury suffered in Wednesday's victory over Carolina. Alexandre Burrows moved up the lineup to skate alongside the Sedins on Vancouver's top line.