The Lightning emerged from a wild second period with a 3-2 lead and Alex Killorn padded that advantage by deflecting a shot past Eddie Lack 7:28 into the third.
Tampa Bay had lost two in a row, but ended that brief skid in the opener of a four-game road trip through Canada.
Valtteri Flippula and Tyler Johnson scored 20 seconds apart in the second period to give the Lightning a 2-1 lead, but Zac Dalpe then tied it with his first goal of the season and the sixth of his four-year career.
Kucherov put Tampa Bay in front with a power-play tally.
Ben Bishop made 28 saves to get the victory. Lack allowed four goals on 33 shots.