Killorn gave the Lightning a 2-1 lead when he redirected Anton Stralman's shot from the point past Senators goalie Robin Lehner. Killorn, who had an assist on Callahan's goal, has an eight-game point streak.
The third period was the difference. The Lightning outshot the Senators 15-10 and scored three times.
"I'm not sure what the numbers are but it feels like we're having a lot more success in third periods," Morrow said. "Pucks are going in the net for us. I don't know if it's wearing teams down or we're getting breaks from hard work. I feel like we're working hard and we're getting rewarded for it."
Ottawa (10-9-4) has lost five of its past seven games.
"We're getting chances but they aren't going in the net and we need goals right now," Senators forward Clarke MacArthur said. "So the positive thing is, you're getting chances and you're still around the net. They need to go in and that's our job to get them in."
Tampa Bay tied the game 5:13 into the second period when Callahan scored his 11th off a pass from Steven Stamkos on a 2-on-1. Stamkos beat Senators defenseman Eric Gryba to a loose puck to start the odd-man rush.
Morrow scored his first goal with the Lightning, in his 22nd game, for a 3-1 lead in the third period. Cedric Paquette took the puck from Senators defenseman Cody Ceci behind the net and flipped a pass to Morrow in the slot, who sent a one-timer past Lehner.
"It was good to get that one out of the way because you start to grip your stick and wonder if it's ever going to happen," Morrow said. "It's a good feeling when they go in. Hopefully they'll be a little easier from now on."
Kucherov's third-period goal came on a power play and was his ninth. Tyler Johnson had his team-high 17th assist.
"We have four lines that can score; we don't have to rely on one or two guys or a line or two," Johnson said. "And Lehner is a good goalie. I played against him a lot in the AHL when we were down there. He's big and athletic and you have to be able to get those shots in traffic and get second opportunities."
Lehner, who made 38 saves in a win against the Lightning on Oct. 11, finished with 28 saves Saturday.
"I think the first two periods we played really good," Lehner said. "I think we kept battling I the third, but details matter in this game and they scored three times. That's what good teams do."