Victor Hedman and Mikhail Sergachev scored, and Andrei Vasilevskiy made 28 saves for Tampa Bay. Nikita Kucherov's point streak ended at 12 games.
"Looking back, we still only scored two goals," Lightning coach Jon Cooper said. "I felt we deserved a little better fate. We hit the post three times. I know it's only three games [on the road], but it's the West Coast and guys can get tired.
"Tonight was just not our night. For 16 or 17 games, it was our night. Give them credit; they skated well. Hitting three posts when you're on a streak, those probably go in. Tonight, they didn't. This is still a great group of guys to coach."
Couture gave the Sharks a 1-0 lead at 9:27 of the first period before Pavelski made it 2-0 at 10:01 with his 25th goal of the season, redirecting a pass from Burns.
Hedman made it 2-1 on a snap shot from the low slot.
Kane's power-play goal made it 3-1 at 4:23 of the second period on a wrist shot from the left circle.
Sergachev scored his first goal of the season 30 seconds later from close range to make it 3-2 at 4:53.
Kane scored his 200th NHL goal at 5:18 of the third period with a wrist shot from the left circle to make it 4-2.
Radil made it 5-2 at 12:20 when he scored on a rebound.
"We're a confident group," Kane said. "We know what type of team we are when we play as a team and we're rolling. I thought we did a good job for the most part defensively."