Patrick Brown made it 3-1 at 6:21 of the second period when he controlled a rebound off a shot from Kastelic and chipped it over Elliott.
Brandon Hagel cut it to 3-2 at 13:38 with a 5-on-3 goal when he stuffed in a rebound off a shot from Stamkos.
Hedman tied it 3-3 at 14:06 with a one-timer from the top of the right circle on a power play.
"Turnovers, we're giving up so many chances again, it's tough to win like that," Hagel said. "Sometimes you just have to simplify. We're not playing smart hockey and it costs us. Obviously it was a tough trip, so it will be good to go back home and hopefully give a better effort."
NOTES: Brady Tkachuk (assist) got his 80th point to join
Tim Stutzle
(86). Ottawa hasn't had multiple players with 80 points since 2007-08 (Jason Spezza, Dany Heatley, Daniel Alfredsson). … Along with his goal, Gauthier had a team-high four hits in 8:08 after missing six games with an illness. … DeBrincat has four goals in the past seven games. … Elliott has given up six goals in each of his past three starts. … Lightning forward Nikita Kucherov had three assists and has 81 this season, his second time reaching 80 (87, 2018-19). … Hedman played his 973rd game to pass Martin St. Louis for third in Lightning history behind Vincent Lecavalier (1,037) and Stamkos (1,001).