Hagel cut it 2-1 at 9:16 of the second period with a shot from the left circle that beat Lindgren stick side. Hagel extended his point streak to 11 games (16 points; eight goals, eight assists).
Anthony Mantha pushed it to 3-1 at 12:14 when he finished off a 3-on-1 rush with a one-timer to the near post.
Paul cut it to 3-2 at 2:42 of the third period after he skated past Capitals defenseman Ethan Bear and beat Lindgren with his backhand.
"We knew we had to do what we could to win that game," Paul said. "And we came on hard and unfortunately fell short. And yeah, it's a tough one, you know? Just got to move past it now and just focus on these next couple of games and bear down."
Motte tied it 3-3 at 3:06 with a wrist shot from the left hash marks.
"Every game is disappointing when you lose," Paul said. "You play 82 games and we try to win them all. Unfortunately, it doesn't happen. You can't get too hung up on it. This time of year it's crunch time. We’ve got to find a way to win games. It wasn't tonight, but you’ve got to be able to turn the page on [it] pretty fast."
Strome had a goal overturned at 4:25 after Tampa Bay challenged for offside.
Capitals forward T.J. Oshie left the game midway through the third period with what appeared to be a noncontact injury. He crawled off the ice and sat on the bench briefly before being helped down the tunnel.
"[The concern] is fairly high," Carbery said. "We'll evaluate. I'll know more tomorrow, but it's never a good sign when you see a player leaving the ice like that."
NOTES: Lightning defenseman Haydn Fleury left the game early in the first period after taking a hit from Wilson along the end boards. Cooper had no update. ... Lightning forward Nikita Kucherov had an assist and leads the NHL with 95 points (36 goals, 59 assists). ... Tampa Bay defenseman Victor Hedman was minus-2 with three shots on goal in a game-high 29:56 of ice time. … Lightning forward Brayden Point had a three-game goal streak end.