Killorn made it 2-0 at 7:40 when he took the puck from Coyotes defenseman Dysin Mayo at the blue line and finished a breakaway with a backhand.
"I think we were fortunate enough to get the quick lead," Killorn said. "It was probably not our best second and third period. I think maybe we were relying a little bit too much on that lead. There's certain things we could do better."
Stamkos extended the lead to 3-0 at 10:41 after he banked the puck off Prosvetov down low. The power-play goal ended an 0-for-15 stretch without injured forward Nikita Kucherov in the lineup.
"Work in progress," Cooper said of the power play. "I liked tonight because we were shooting the puck. We put the puck in a lot of good areas, but it is a work in progress. When you lose a player like [Kucherov] for a couple of months it's going to have an effect. It changes the role of some of the players and it just takes time."
Alex Barre-Boulet made it 4-0 at 13:38 of the second period when he chipped in a pass from Mikhail Sergachev to the slot.
Crouse cut it to 4-1 at 12:36 of the third period after video review determined his shot crossed the goal line and hit the camera in the net.
"We've got to tough it out," Crouse said. "It's up to us. We need the will to win. No one feels sorry for us. It's good to have a back to back [at the Washington Capitals on Friday]. We don't have a day or two to dwell on it. Everyone's a little bit frustrated. We have to stick to it, put this one behind us and move on."
Andrej Sustr scored at 18:23 for the 5-1 final. His last NHL goal was Dec. 5, 2017 against the New York Islanders.
NOTES:Killorn has scored four goals in a three-game goal streak. His assist was the 200th of his NHL career. … Stamkos has four multipoint games this season. … The Lightning had scored just one first-period goal this season coming into Thursday. ... The Coyotes have given up 10 power-play goals on 18 opportunities (44.4 percent) to rank last in the NHL. ... Coyotes forward Clayton Keller played his 300th NHL game.