Killorn, who was with center Anthony Cirelli and Taylor Raddysh, made it 3-3 at 16:13 on a deflection from the slot.
Tampa Bay then scored three goals in a span of 3:04 of the third.
"I felt like we just kind of gave up once they tied the game," Devils forward Tomas Tatar said. "We were not playing the same way as we did in the first half of the game."
After Hedman's goal, Kucherov scored on a breakaway at 9:14 for a 5-3 lead, and Bellemare made it 6-3 at 10:21, resulting in Gillies being pulled.
"We had to get a kick in the butt and finally we woke up," Bellemare said, "but I would put that one in the unacceptable category."
NOTES: The Lightning have four multigoal comeback wins this season, third-most in the NHL behind the Minnesota Wild (six) and Columbus Blue Jackets (five). ... Hedman, who played a game-high 24:30, leads NHL defensemen with 50 points (11 goals, 39 assists). ... Cernak had three shots on goal and five hits in 18:44 after missing nine games with a lower-body injury. ... Tampa Bay has eight days until its next game, against the Edmonton Oilers on Feb. 23; New Jersey's next game is in nine days, at the Pittsburgh Penguins on Feb. 24. ... It was the first three-point game and seventh multipoint game of Mercer's NHL career. He has played 50 games, all this season.