The United States women’s national team under new broom Emma Hayes are two wins from a record-extending fifth Olympic gold medal, after a Trinity Rodman wonder strike in extra time broke open a tightly contested quarter-final with Japan that failed to catch fire until the final reel.

Related: The Emma Hayes effect: reborn USWNT look like Olympic contenders again

The women’s football tournament at these Paris Games should at least begin to determine whether the Americans’ torpid 46-month run under the departed Vlatko Andonovski will be remembered merely as a flop period or, as a practically salivating global football establishment have openly manifested, a permanent regression from the class of perennial favorite.

The jury remains out but early returns are promising. Hayes, the former Chelsea manager who took the reins two months ago, has guided the US to four wins in her first four meaningful games. Held to only four goals in as many games during last year’s World Cup washout, the Americans smashed in nine on a perfect run through the Olympic group stage. The wait for a 10th on Saturday afternoon was an often uncomfortable one, but its arrival brought Hayes’ squad within touching distance of the podium.

France's hopes of home Olympic gold ended in heartbreak as they lost 1-0 to Brazil in their quarter-final match in Nantes.

The hosts missed a golden opportunity to take an early lead when Sakina Karchaoui's penalty was saved by Brazil keeper Lorena. Griedge Mbock Bathy then struck the crossbar with a header, before Lorena denied Delphine Cascarino in a one-on-one just before half-time.

Brazil started the second half stronger but France fought back and the winner seemed a matter of time as Selma Bacha and Cascarino missed chances. But in the 82nd minute, France failed to clear Adriana's shot and Gabi Portilho [pictured] slotted home to give Brazil a lead they held through 16 minutes of stoppage time.

The world champions, Spain, await in the semi-finals after they fought back to deny Colombia a surprise victory. The underdogs had taken a 2-0 lead through Mayra Ramírez and Leicy Santos, but Jenni Hermoso pulled one back in the 79th minute and Irene Paredes levelled in the seventh minute of injury time.

With no goals in extra time, the match went to penalties and it was Colombia who blinked first, with Spain keeper Cata Coll keeping out Catalina Usme’s weak attempt. Colombia’s Liana Salazar also missed her spot kick and Aitana Bonmatí coolly converted Spain’s fourth spot-kick in a row to send them through.

Germany will face the USA in the other semi-final after they ended Canada's tumultuous title defence on penalties after a goalless draw. Goalkeeper Ann-Katrin Berger was the hero, saving from Ashley Lawrence and Adriana Leo before stepping up to score the winning spot-kick. Reuters

“It was very tactical,” Hayes said. “That’s why I liked it so much. It was a match of grit and determination. It was decided by one moment of brilliance.”

The opening half was a dour affair that saw the US dominate possession but remain unable to generate meaningful chances against Japan’s well-organized block. Playing behind a friendly crowd that broke into chants of “U-S-A!” every few minutes, the Americans took a more thoughtful approach instead of just ramming it down the park as they might have in the not-so-distant past, but their patience was nearly punished twice in the 10 minutes before the intermission as Japan created two moments of genuine danger through Kiko Seike and Miyabi Moriya.

At that point, signs of fatigue among the US attackers, none more than Rodman, were putting Hayes’ decision not to rotate players during the condensed group stage into a harsh glare. Elsewhere on the pitch the captain, Linsdsey Horan, a one-time Paris Saint-Germain midfielder back on familiar ground, was a step slow. Korbin Albert, Hayes’ one change from Wednesday’s win over Australia, was ineffective even before staying on through a knee injury near the end of regular time.

It didn’t get much better after the intermission. By the hour the US looked even more lackluster and bereft of attacking ideas while the Nadeshiko, despite a lack of genuine scoring opportunities, seemed more likely to break through. They received an immediate boost when West Ham’s Riko Ueki came on for Mina Tanaka after 70 minutes, putting a shot directly across the mouth of the goal, inches outside the post.

The Japan manager, Futoshi Ikeda, had long since introduced three substitutions by the time Hayes made her first switch ahead of extra time with Lynn Williams, the all-time NWSL goal scorer, coming on for Mallory Swanson. By then exhaustion on both sides was leading to defensive lapses, most notably when the Japanese keeper, Ayaka Yamashita, was forced to make a point-blank save after Smith took one touch too much many.

The Americans desperately needed a moment of inspiration to avoid a penalty shootout and they got it in the dying moments of the first extra-time period from Rodman, who ran into space behind Japan’s high press on to a long pass from Crystal Dunn, cut back across to her left to beat left-back Hikaru Kitagawa and curled a left-footed thunderbolt into the top corner past the fully extended Yamashita for her third goal of the tournament.

“I have flashes of what happened,” the 22-year-old Washington Spirit forward said. “I got into the area, I received a beautiful pass, then I took a shot. It wasn’t that I didn’t have confidence in my shot, but at the same time, I couldn’t believe it.”

Now it was the Americans’ turn to sit back in a block and kill off the game and after a mostly straightforward quarter-hour, Hayes’ team could fête their progress to the last four before a rollocking crowd that included former USA stars Megan Rapinoe and Mia Hamm and the rapper Snoop Dogg.

The US advance to the semi-finals, where they will face Germany, who eliminated Canada on penalties, in Marseille.