SAINT-DENIS, France — Before their decisive match against Austria, Iceland took to the field at the cavernous Stade de France in their suits, their polished dress shoes pressing into the grass. They stood together in small bunches, their hands in their pockets, looking up mostly, taking in the surroundings of giants. Some of them walked over to the stands to greet the first of the Icelandic fans who had filed in. They took pictures and signed autographs. A chubby-cheeked boy reached out for a hug from Johann Gudmundsson, putting his head on the shoulder of his hero and closing his eyes.

They looked as though they were savoring the afternoon, players and fans alike, knowing full well that a loss would have sent them home, back to their beautiful island. They hoped. Some of them might have even believed. But they could never have imagined how the next few hours would unfold. Iceland would win a frantic, emotional game 2-1 and they would advance to play England in Euro’s Round of 16 on Monday.

It will be the latest in an ever-lengthening string of the biggest games of their lives.

“It was incredible,” Kari Arnason, the Man of the Match, said after. “Unbelievable. We’re going to play England.”

In the 18th minute, Iceland struck. Aron Gunnarsson made a long throw from the right sideline — replays showed that his left foot went over it — and Arnason advanced it deep into the box with a glancing header. Jon Dadi Bodvarsson took over from there. Just outside the six-yard box, he pulled a low shot across his body, past a frozen Robert Almer and into the goal.

Austria kept by far the majority of the possession, however, and they came close to equalizing again and again. They were never closer in the first half than when they were awarded a penalty in the 36th minute. Ari Skulason pulled down David Alaba by his arm. Aleksandar Dragovic lined up to take the kick.

He drove the ball into the base of the post.

At last, in the 60th minute, Austria’s fadeless attack paid off. Substitute Alessandro Schopf made a long, twisting run starting from well outside the box. He cut through the waning Iceland defense — they were looking so tired even then — and pulled the ball beyond the reach of a diving Hannes Halldorsson. It bounced and bobbled into the net.

The fate of both teams now hung on the balance of a single goal. Iceland would survive with the draw; Austria needed the win. They were obviously going to press like weight. Iceland were about to endure the longest 30 minutes in their history, which is really saying something given their winters.

“I started seeing stars,” Gunnarsson said. “But that’s our Viking spirit. We keep on fighting to the end.”

The scoreboard flashed with almost unbelievable updates from the wild match between Hungary and Portugal, taking place at the same time in Lyon. Their relentlessness only added to the sweaty, gut-sick feeling inside the Stade de France, where shared fates rose and suffered by proxy. It was the hottest day in Paris since the beginning of the tournament, and the Icelanders must have felt as though they had been put on a long, slow boil. With every passing second they were looking more and more spent.

Iceland played the same sort of match against Hungary, buckling in the end. They nearly did again. Schopf almost claimed his second, his close-range shot clattering off Halldorsson’s long legs. Arnason had already made two desperate blocks by then, the ball once deflecting off his foot and over the bar.

Now four minutes of injury time. An eternity.

In the dying seconds, the hundredth Icelandic clearance rolled slowly toward half. Austria, nearly all of Austria, were massed at the wrong goal. (Even Almer had been pushing forward during a corner only moments before.) Elmar Bjarnason picked up the ball and began an epic, stumbling run. Chased by a pair of teeth-gritting Austrians and two of his own gasping teammates, he looked as though he wasn’t going to make it into the box without collapsing first. He slid the ball left, across to Arnor Ingvi Traustason, who punched it past Almer.

Iceland had done it.

“Probably the hardest, most nerve-wracking 30 minutes I’ve ever played,” Gylfi Sigurdsson said. “But the last 10 seconds, it was all worth it.”

The Austrians, eliminated, fell to the ground. The Icelanders somehow stayed on their feet and staggered like marathoners toward their soaked-through supporters. When they finally did fall, they fell instead into the arms of each other, savoring one of those rare moments when their waking lives eclipsed even the dreams of chubby-cheeked boys.

“I think we are going to tell them just to go to sleep,” co-coach Heimir Hallgrimsson said when asked what he will say to his players.

But why would they ever want to close their eyes again?