Hagel put Tampa Bay back in front 2-1 at 18:03 of the first on a rebound after Point's initial shot hit the crossbar.
"He's been getting his chances, they just haven't been going in," Cooper said of Hagel, who didn't have a point and was minus-2 through the Lightning's first five games. "That's why you're fired up when they go in for him. He's played well, he works, he retrieves pucks, he does all these things. He's had his chances."
Perry extended the lead to 3-1 at 3:12 of the second period, finishing a cross-crease pass from Pierre-Edouard Bellemare at the left post.
Beauviller cut it to 3-2 at 9:36 when his backhand on a rush trickled through Elliott.
"Much better effort. (We) probably deserved better, but a step in the right direction," Islanders forward Cal Clutterbuck said. "We just made some big mistakes at some bad times."