Bellemare then made a beautiful, backhand saucer pass to Stamkos, who was open in the slot and ripped home his 28th goal of the year into the top right corner just 20 seconds after his penalty expired.
Taking the 2-1 lead into the third period, the Lightning found themselves in a tie game once again when Timo Meier scored with 10:19 remaining in regulation.
With Jonas Siegenthaler in the penalty box for interference, the Bolts regained the lead at the 15:13 mark of the third. When Nikita Kucherov sent a crafty backhand pass up to the point for Sergachev, he put the puck on a platter for Stamkos, who rifled home his patented one-timer to the far post to give Tampa Bay a 3-2 lead with 4:47 remaining.
After missing the previous game with a lower-body injury, Stamkos looked terrific on Thursday night with two goals and an assist. Not only did he look better, but he felt much better as well.
"I probably forced that game against Winnipeg the other night," Stamkos said. "I probably should've sat that one out.
"It's that time of the year where obviously if it's a playoff game, you want to battle through some things, but you want to be feeling as good as you can coming down the stretch.
"I think it was probably the smart decision not to play the other night. Get a couple treatment days and an off day, then skate yesterday. I felt 10 times better tonight than I did against Winnipeg the other night, so it was well worth it."
As Tampa Bay looked to close out the game, Nick Paul was sent to the penalty box for tripping and New Jersey pulled Schmid for a 6-on-4 advantage before Meier buried his second goal of the night to force overtime.
After no goals were scored in the extra frame, Ross Colton and Alex Killorn found twine in the shootout and Andrei Vasilevskiy stopped two-of-three shots to earn the second point for the Lightning.
"We knew coming into this kind of back-to-back almost here, we were playing a really good team and it was going to be a good test," Stamkos said. "We wanted to see where we were against a team that's having a great year.
"Two in their home rink. We hadn't been playing very well on the road. We could feel our game getting a little better over the course of the last week and a half here.
"Those two games were, I think, a byproduct of us playing the way we want to coming down the stretch and into the playoffs."