Hedman scored twice, Sergachev had a goal and two assists and Cal Foote had a goal and an assist for the Lightning (37-12-6), who began a six-game road trip. Andrei Vasilevskiy made 24 saves; he is 12-0-0 in his NHL career against the Blackhawks.
"It's going to say Chicago scored three goals on [Vasilevskiy]," Tampa Bay coach Jon Cooper said. "That was one of the more remarkable goalie performances I've seen in a long time. There's making big saves, but watch the highlights tonight of saves he made. That's what kept us in it. Our work ethic is what propelled us to win it. We didn't have that going at all in the first period."
Patrick Kane, Alex DeBrincat and Dylan Strome each had a goal and an assist for Chicago (20-29-8), which has lost three of its past four games. Marc-Andre Fleury made 26 saves.
"First period, I thought we were fine," Blackhawks coach Derek King said. "We played a good period. Second period too, I think if we score early, maybe it changes things, and then they come right back and that's what they do. They strike and they strike fast, and it was just bang, bang, bang. Tough for us to climb out of that. We don't have that depth of firepower."
Strome scored at 7:51 of the first period on a one-timer from the right face-off circle to give Chicago a 1-0 lead.
Taylor Raddysh tied it 1-1 at 1:09 of the second period when he redirected Foote's backhand.
Kane gave the Blackhawks a 2-1 lead at 6:09 with a wrist shot from beyond the circles.
Hedman tied it 2-2 at 15:17 on a one-timer from Steven Stamkos, and Foote gave Tampa Bay a 3-2 lead 1:31 later at 16:48 off a pass from Anthony Cirelli at the left side of the net. It was Foote's first goal of the season.
"Anytime you can add to the offense, it's nice," Foote said. "Credit to [Nikita Kucherov], [Ryan McDonagh] and [Cirelli] on that play. It was all of them, and I was in the right spot at the right time."