The Lightning were 0-2-1 at Amalie Arena before beating the Coyotes, who fell to 0-6-1 and have lost the first two games of their six-game trip.
Steven Stamkos had a goal and an assist in the first period, Andrej Sustr also scored and Andrei Vasilevskiy made 24 saves. He lost his shutout bid when Lawson Crouse scored with 7:24 remaining in the third period.
The Lightning scored three times in the first period after managing one goal in the opening period of their first six games.
''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.''
Rookie goalie Ivan Prosvetov, playing his first game since being recalled from Tucson of the American Hockey League on Wednesday, finished with 23 saves for Arizona in his fourth NHL game.
Tampa Bay took a 1-0 lead 5:39 into the game when Killorn finished off a 2-on-1 break by taking a pass from Stamkos and beating Prosvetov from inside the left circle. Killorn made it 2-0 at 7:40 with his sixth of the season, scoring on a breakaway after Coyotes defenseman David Mayo misplayed the puck at Tampa Bay's blue line.
It became 3-0 at 10:41 when Stamkos ended Tampa Bay's 0-for-15 power-play drought. He controlled the puck behind the net to the right of Prosvetov and backhanded it off the goalie's back and into the net for his fifth goal.
It was the Lightning's first power-play goal at home this season in 12 opportunities.
''The first goal happened, and then the second goal and the third, and before you know it you find yourself chasing one of the best teams in the league,'' Crouse said.
Alex Barre-Boulet scored his first of the season at 13:38 of the second period, using a backhand chip to deflect Mikhail Sergachev's left-point pass over Prosvetov.
The Coyotes carried the play for much of the final two periods, but had nothing to show for it until Crouse scored his second of the season, beating Vasilevskiy from the right circle.
''We knew it would be a hard task down 3-0,'' Coyotes coach Andre Tourigny said. ''I liked the way the guys showed up in the second period. We created a lot of offense against one of the best teams in the league.
Sustr scored his first of the season with 1:37 remaining on a slap shot from inside the blue line.
Lightning coach Jon Cooper was happy with the result, if not the full 60-minute effort.
''We did what we had to do,'' Cooper said. ''Was it a perfect game for us? By no means, but when you have a desperate team like Arizona, just getting the lead was imperative for us and the guys did what they had to do. We'll just take the two points and move on.''
The Lightning led after the first period for the first time in their seven games this season; the Coyotes have yet to take a lead into the dressing room after the opening period. . Sustr's goal was his first in the NHL since Dec. 5, 2017, when he scored for the Lightning against the New York Islanders.
The Coyotes have allowed 10 power-play goals on 18 chances, a penalty-killing rate of 44.4% that's last in the NHL. Their power play went 0 for 3 and is 2 for 19 (9.5 percent).
Arizona has been outscored 15-2 in the second period this season. Killorn has goals in three consecutive games.
Coyotes: Play second of back-to-back games at Washington on Friday night
Lightning: Host Washington on Monday night.
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