Sharangovich put the Devils ahead 4-3 at 9:34 with his second goal of the game.
Brayden Point was awarded a penalty shot at 10:06 after he was pulled down by Ryan Graves, but his shot went off the crossbar.
Tatar scored an empty-net goal at 18:44 to extend the lead to 5-3.
"We're going to be down in games and we have to find a way to get back in them, find a way to win them, find a way to get points," Devils coach Lindy Ruff said. "I knew going into tonight that if we got the opportunities, we were going to have to finish to be a good team."
Rutta gave the Lightning a 1-0 lead at 5:49 of the first period with a wrist shot from the top of the right circle.
"I think we were lacking a little bit of urgency the whole game and it bit us at the end," Rutta said. "Normally we play with a little more grit. I think we were lacking that tonight."
Sharangovich tied it 1-1 at 13:56 of the first with his first goal of the season, a snap shot from the low slot.
"He's been playing great for us," Blackwood said. "Just been a little bit of a snake bite there, but I feel like now that he's got those goals, the floodgates are going to open for him."