A very tough lesson was learned, and with it the most precious of opportunities may have been lost.

The upstart Islanders outplayed the top-seeded Penguins yesterday afternoon at a raucous Nassau Coliseum, but a handful of mistakes cost the home team dearly. Behind three power-play goals, the Penguins won Game 3 of the first-round series, 5-4 in overtime, to take a 2-1 lead in this best-of-seven contest.

“Special teams, they had the edge,” Islanders coach Jack Capuano said. “That’s how they won the game.”

The back-breaker came 8:11 into overtime. Isles defenseman Brian Strait overplayed Sidney Crosby along the side boards, and when the Penguins superstar skated by him and toward the net, Strait had little choice but to grab Crosby by the shoulder and draw a holding penalty.

“I blame myself more than anything,” Strait said. “If you lose body position on a guy like that, your natural reaction is to get in his way, and no matter what the call was, that’s the game.”

On the ensuing power play, Chris Kunitz took a great pass from Crosby and buried his second goal of the game, over the left shoulder of Evgeni Nabokov, who crumpled to the ice face-first.

“The difference is our penalty kill or their power play — I don’t know which one is too good or which one is too bad,” Nabokov said. “That’s where the difference is.”

Coming off a shocking 4-3, Game 2 win in Pittsburgh on Friday night that tied the series 1-1, the Islanders rode their momentum right into their aging hockey barn packed to the rafters with rambunctious fans. They got two goals in the first 5:41, from Matt Moulson and Casey Cizikas, and it all seemed so real — the chance for the Islanders, in their first postseason since 2007, to win their first home playoff game in 11 years.