PARIS -- After a long, hot, ponderous afternoon of frustration, Trinity Rodman took the ball down, shimmied to her left and delivered a brilliant laser when the United States women's national team needed it most.

Rodman's blistering shot from a tight angle rippled the net in the 106th minute, sending the crowd -- including USWNT legend Megan Rapinoe -- into hysterics as the white shirts of the 22-year-old's teammates piled up on top of her.

It was a stunning goal, and one that proved to be the difference in a 1-0 U.S. victory over Japan in a tense quarterfinal match at Parc des Princes on Saturday, meaning the Americans will now face either Canada or Germany in the semifinals on Tuesday in Lyon.

"It was the best moment of my career," Rodman said afterward. "I couldn't have asked for anything better."

The victory guarantees the U.S. an opportunity to play for a medal -- no small thing after last summer's disappointing exit in the round of 16 at the Women's World Cup -- but for much of the match on Saturday, it seemed as if the Americans might not advance at all. They struggled to unlock the Japanese defensive stance which was disciplined, regimented and stifling, only to find a bit of magic just before the extra time break.

Crystal Dunn began the decisive move with a crossfield pass that allowed Rodman to break down the flank. She cut her way into the box, darting past Hikaru Kitagawa, before unleashing a vicious shot with her left foot that whizzed past Ayaka Yamashita's helpless dive.

It was the latest goal scored by the U.S. in an Olympic knockout game since Alex Morgan's late winner at Old Trafford in the semifinals of the 2012 Games, and it allowed the Americans to avoid the lottery of what had felt like an inevitable penalty shootout.

"Was it a game that was most likely to head to penalties? Yeah, absolutely," said U.S. coach Emma Hayes, who repeatedly praised Japan's skill in sticking to its tactics.

"Their block is the best in the world," Hayes added. "They brought out our best patience, which was the most needed skill today ... [Rodman's shot] was a world-class finish."

Playing their match in the same venue where the U.S. men's team crashed out of the Olympic tournament to Morocco a day earlier, the atmosphere -- which had been so heavily against the U.S. on Friday -- was very much behind the Americans from the start.

Through much of the match, though, there were long passages of play where the stadium grew almost silent as the Japanese set up in their defensive posture and the U.S. probed without success. The Americans had the ball constantly; they attempted 1,026 passes on Saturday, according to ESPN Stats & Information, the most by any team in a game in either the past three Olympics or past four Women's World Cups.

Gaudy numbers, to be sure, but without a goal, it left everyone connected to the U.S. -- in the stands and on the field -- feeling a rising worry, especially as Japan had chances, too.

Mina Tanaka shot straight at Alyssa Naeher in the first half, Dunn was forced to clear off the line in the second and Hinata Miyazawa blasted into the side-netting late, ratcheting up the pressure on the U.S. even more.

"I think that we just keep going," forward Mallory Swanson said. "I think that's the mentality that we have. We just keep going. If we miss a chance, we just keep going and we keep trying something different."

They did that, attempting to infiltrate the Japanese end by any way possible, but it was certainly a slog.

The U.S., which was playing without suspended defensive midfielder Sam Coffey, didn't make a substitution until the 90th minute, raising questions from some about potential fatigue.

Hayes bristled at that suggestion in the post-game mixed zone. "I'm paid to do this job, so all I'm interested in is what we do in house," she said. "I don't believe we'd have gone through if we'd have made too many changes."

She added that her belief is that the connections the players are building within this still-developing system are more significant than any tiredness, and there is no denying that that link was on display in the endgame.

After all, Dunn found Rodman. A shooting lane was created. And an afternoon of exasperation transformed, all of a sudden, into one full of utter jubilation.

"That," Hayes said, "was a beautiful goal."