Cirelli put Tampa Bay ahead 3-0 at 1:53 of the second period.
"[Cirelli has] been great, obviously he's playing on special teams now too, on the [penalty kill] and taking big face-offs," Lightning defenseman Victor Hedman said. "With three points tonight, he's been a great fit, so once again that shows the depth we have in the organization with guys coming up from Syracuse [of the American Hockey League] and doing a tremendous job."
Spooner scored to make it 3-1 at 7:10 of the second period. He has 10 points (two goals, eight assists) in six games since he was acquired in a trade from the Bruins on Feb. 25.
The Lightning went ahead 4-1 on Kunitz's goal at 12:35 of the second, and Johnson made it 5-1 at 17:30.
Zuccarello scored at 3:27 of the third to make it 5-2, and Hayes made it 5-3 at 17:41.
Rangers goalie Henrik Lundqvist allowed four goals on 31 shots before being pulled for Alexandar Georgiev with 7:25 remaining in the second period. Georgiev made 13 saves.