Kucherov, who reached 1,000 NHL points on Saturday in a 4-3 win against the Anaheim Ducks, had what appeared to be the go-ahead goal taken off the board at 13:38 of the third period after Vegas successfully challenged for offside.
Hagel also scored for the Lightning (3-4-2), who have won two straight for the first time this season. Andrei Vasilevskiy made 18 saves for his first win of the season in six starts (1-3-2).
"Adversity is a cliche because everyone will say, 'Oh my god, we needed that adversity,’" Tampa Bay coach Jon Cooper said. “Well adversity [stinks] when you're in it. It really [stinks] and you have work your way out of it and it's so much easier than done. We deserved some points out of this game, and we were fortunate to get two."
William Karlsson scored for Vegas (5-1-3), which was coming off its first regulation loss of the season at the Florida Panthers on Saturday (3-0). Lindbom made 26 saves in his NHL debut.
"[Lindbom] kept us in the game,' Karlsson said. "A heck of a game, good kid and apparently a really good goalie. Unfortunately, we couldn't get the two points."
Hagel scored his first goal of the season to give the Lightning a 1-0 lead at 4:53 of the first period, burying the rebound of Victor Hedman's point shot.
"It felt good. It was nice because I still feel like maybe I've got it," Hagel said. "Nah, it's nice to get that one out of the way and then you continue to play hockey. Obviously, at the start of the season it's going to creep in the back of your head, it's human nature. Once that happens, you kind of find your game and you're not gripping it as tight. You're out there having fun and playing hockey like you have your entire life."
Karlsson tied it 1-1 at 18:25 of the first when he whipped a wrist shot from the left circle past Vasilevskiy on the short side.
"It wasn't the start we wanted, we couldn't get in a rhythm. We got through the penalties and started breaking pucks out better and then couldn't get through the neutral zone, so it was like a domino effect," Vegas coach Bruce Cassidy said. "But you know, you're tied going into the third. It's not easy to win in this league. We recognized that and talked about that. We just have to manage the puck better in overtime with only three guys on the ice."