With Panthers forward Derek MacKenzie in the penalty box for slashing, Quine scored on a one-timer from the top of the right circle after taking a feed from defenseman Marek Zidlicky. The puck bounced in off the right post.
"It was opening up for a one-timer, and [Thomas] Hickey made a good play to Zidlicky and he put a nice fake on it and put it right in my wheelhouse and I just ripped it," Quine said. "I put it on net and near the post and saw everyone celebrating a little bit. It was nice for it to go in."
The Islanders won a Game 5 for the first time since 1987, snapping a streak of 11 defeats. The team that wins Game 5 in a series tied 2-2 is 189-52 (78.4 percent), according to Elias Sports Bureau.
The Panthers lost their sixth consecutive overtime playoff game, a streak that includes Game 3 of this series (4-3 on Sunday).
"It's a tough loss," Florida coach Gerard Gallant said. "We played a great game again tonight. The mood is a little bit down, but by the time they get home and they get something to eat, they'll be ready to go back and go to New York and have a great game."