Brock Nelson scored for the third straight game for the Islanders (28-27-9), who had won six in a row at home.
Ilya Sorokin made 18 saves but left after the second period with an upper-body injury. Semyon Varlamov made 12 saves in relief.
"It was a pretty even game," New York coach Barry Trotz said. "There wasn't a lot of chances both ways. Both teams, I think, were playing with a lot of bite and a lot of detail. There wasn't much room.
"They had a little bit of a push, but we were fine the first probably eight, nine minutes (of the third). They got the third, and honestly the third goal sort of sucked the air out of our bench for a while."
Nelson gave the Islanders a 1-0 lead at 18:14 of the first with a wrist shot from the right face-off circle on a 2-on-1 with Anthony Beauvillier. It was Nelson's seventh goal in six games.
Hedman tied it 1-1 at 1:10 of the second, scoring on a wrist shot from the right circle during a power play.
Killorn put Tampa Bay ahead 2-1 at 9:47 on a redirection of Anthony Cirelli's feed.
"We could have been up a couple early," Lightning coach Jon Cooper said. "We weren't, and then we fall behind. When things aren't going as well as you think they can and confidence becomes an issue, that's when you have to strap the boots on and go back to work, and they did that.
"Nobody was hanging their heads. That was a big thing for us and (a) good result for us tonight."