The Predators led 2-1 with 7:05 remaining in the third period, before Hurricanes forward Martin Necas scored a wraparound goal to tie it 2-2.
Carolina forward Jordan Staal knocked the puck out of the air for a 4-on-4 goal 2:03 into overtime.
"It was fun. Overtime, obviously, at home, especially with what we've gone through the past couple games," Staal said. "To get a big one like that felt really good, but we've still got some work to do."
Forward Yakov Trenin, who scored the Predators goals in Game 5, said they need to be relentless in Game 6.
"Play as we've played in home games," he said. "Always put pressure on them, don't let them get out of the zone. Put them on their heels. Just attack, attack."
Nashville won Games 3 and 4 in double overtime after losing Games 1 and 2. The Predators were next to last in the Discover Central Division on March 15 then went 18-6-1 in their final 25 games to qualify for the Stanley Cup Playoffs as its No. 4 seed.
They won Games 6 and 7 against the Anaheim Ducks in the 2016 Western Conference First Round.
"A couple of adjustments, but I know our guys will be ready to go," Hynes said. "You take the things out of this game. There are certainly some things we did really well, certainly things we can do better. The thing I like about the group, as you've seen over the last 2 1/2 months, when we're in situations like this, is our ability to refocus, put it behind us, get better."