Sebastian Aho and Brent Burns each had a goal and an assist, and Teuvo Teravainen scored for the Hurricanes (9-6-0), who have won six of their past eight games (6-2-0) but were coming off a 5-2 loss at the Florida Panthers on Friday.
"It’s what we were looking for right from the start of the game," Brind'Amour said. "We got what we deserved tonight. I think we were on it and ready to go from the start of the game."
Jonas Johansson made 28 saves for the Lightning (6-5-4), who have lost two straight and are 1-2-1 in their past four.
"Last two times we've played these guys we shut these guys out, so there's a little give and take," Tampa Bay coach Jon Cooper said. "For us, though, to come home for just two home games and go back on the road, we've got to get these points. So, it's disappointing to come out of these two games without points."
The Lightning were without forward Nikita Kucherov, who was a late scratch because of an illness. Kucherov entered the game second in the NHL with 23 points (11 goals, 12 assists) in 14 games.
"He's one of the best, if not the best, especially on the power play," Tampa Bay captain Steven Stamkos said. "We weren't prepared for that coming in. It was kind of a last-second thing. Guys stepped up and did their best, but we couldn't get it done. Can't replace a guy like ‘Kuch.’"