Nicholas Paul nearly tied it when his shot hit the crossbar with 3:18 remaining in the third period. Palat scored five-hole on a breakaway to make it 4-2 with 1:41 left.
"It was a big thing for us to come out in the third and not change, not chase the game, even with it tied," Lightning coach Jon Cooper said. "We didn't and got rewarded for it."
Cirelli scored into an empty net to make it 5-2 with 34 seconds remaining.
"In those last two periods, we didn't give very much up," Cooper said. "If you want to win hockey games in this league, you've got to defend. And we're used to scoring goals, we have been over the years. Taking [injured forward Nikita Kucherov] out of our lineup probably takes away half a goal, maybe a goal a game, because that's what he can deliver for you."
Logan Shaw tipped a point shot to make it 5-3 with 16 seconds remaining.
"We played a very mature hockey team and they just kept playing their game for the whole 60 minutes," Tkachuk said. "I think we can learn from their team. They're two-time defending Stanley Cup champions and work to the very end."
NOTES:Elliott, who won for the first time with the Lightning, has at least one win with six NHL teams (Philadelphia Flyers, Calgary Flames, St. Louis Blues, Colorado Avalanche, Ottawa Senators). … The Senators were 0-for-4 on the power play and are 1-for-13 in their past four games. … Lightning defenseman Mikhail Sergachev served the first of a two-game suspension for an illegal check to the head of Toronto Maple Leafs forward Mitchell Marner in a 2-1 overtime loss Thursday.