Dzingel put the Senators ahead 5-4 with a wrist shot at 10:30, and Stone made it 6-4 with 12 seconds remaining.
"I thought [Anderson] did a really good job in the third of shutting the door," said Senators forward Matt Duchene, who had a goal and two assists. "I'm really proud of the boys, and the character we showed tonight is the character that we have in here."
Joseph scored at 8:49 of the first period to get the Lightning to within 2-1. Hedman beat Anderson from the slot to tie it 2-2 39 seconds into the second.
Braydon Coburn scored at 9:15 to make it 3-2, and Joseph scored his second of the game at 14:32 to give the Lightning a 4-2 lead.
"It's a tough loss," Joseph said. "We have to look at what we did bad in the third period. I think we have to help [Vasilevskiy] a little bit more."
Duchene gave the Senators a 1-0 lead on a wraparound goal 4:58 into the game. Ryan scored on the rebound of a Cody Ceci shot at 7:21 of the first to make it 2-0.