The hype around the US at this World Cup finally appears to be waning, even as the Americans beat a resilient Spain side 2-1 on Monday with the help of two penalties. Both spot kicks were scored by Megan Rapinoe to put the Americans through and negate a mistake in the American back line that gifted Spain their goal. The reigning champions now move on to the quarter-finals, where they will face hosts France on Friday in a clash of arguably the two strongest teams left in the tournament.

“We showed a lot of grit and experience in this game,” Rapinoe said. “As we get into knockout rounds, it’s more stressful, there’s more pressure, the games are more intense. It was important for us to stay in it.”

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It took just seven minutes for the Americans to open the scoring, and it initially looked like they were on their way to another dominant victory. Spain midfielder Mapi León took Tobin Heath down in the box, and Rapinoe calmly stepped up to slot the penalty home.

But just a couple of minutes later, the Americans were culpable in blowing their lead. Alyssa Naeher came well out of her goal and passed the ball to Becky Sauerbrunn, who was dispossessed by Spain forward Lucia Garcia. The toe-poked ball ended up at the feet of Jenni Hermoso, who fired past an out-of-position Naeher.

“Blindside I got pressed and I tried to play my No6 when I should’ve put my foot through it anywhere to clear it but I tried to play out of it,” Sauerbrunn said, taking the blame. “She got a nick on it and then Hermoso got it – good finish.”

Facebook Twitter Pinterest Jennifer Hermoso equalised for Spain from the edge of the USA box. Photograph: Marc Atkins/Getty Images

The blame for the goal lay with both Naeher, who should have picked out a better pass, and Sauerbrunn, who needed to be aware of the pressure on her and move the ball more quickly. But Naeher’s error was in keeping with concerns going into this tournament that the Americans have failed to discover an heir to Hope Solo. Naeher had a gaff in the USA’s game against Chile as well, but the goal conceded then was ruled offside.

The players have defended Naeher at every turn, however, and after the game Sauerbrunn again dismissed any criticism of her goalkeeper, even as she stopped short of saying it was unfair for the media to raise doubts about Naeher.

“When we have an attack a potent as ours, you have to find a narrative so why not have it be the untested backline or the untested goalkeeper?” Sauerbrunn said. “I don’t know if it’s unfair – it’s just how soccer is. Alyssa has come into this tournament without any World Cup or Olympic experience, but she’s shown time and time again that she’s an excellent goalkeeper.”

The Americans found the game winner in the second half, however. And, in keeping with the theme of this World Cup, it involved some VAR controversy. In the 71st minute, Virginia Torrecilla made contact with Rose Lavelle but after a lengthy review, the VAR team gave the referee the chance to review the incident at the pitch-side monitors. She saw nothing to persuade her to reverse her decision, however, and Rapinoe scored an almost identical penalty to her first.

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For Spain, it was a heartbreaking end to their second-ever Women’s World Cup after being knocked out in the group stage four years ago. Despite losing to the US, the Spaniards had arguably their best game of the tournament. “I actually think we deserved more, but sometimes it’s like that,” said Spanish midfielder Vicky Losada. “I’m so proud of the team.”

Coming into the match against Spain, the Americans had looked unstoppable, scoring 18 goals and conceding none in the group stage. But Thailand and Chile are minnows compared to the No 1-ranked Americans, and Sweden fielded a mostly second-string team.

USA manager Jill Ellis went with her first-choice starting lineup against Spain, except for midfielder Lindsey Horan, who was held out in place of Samantha Mewis. US Soccer called it a “coach’s decision” and said Horan wasn’t injured. But Horan is on a yellow card, and would have missed the quarter-final if she had been booked again. It was a deserved start for Mewis, who has been very good in the minutes she has played in this tournament so far. But the decision to omit Horan could be viewed as overconfidence on the part of Ellis.

Of course, Ellis has shown a willingness to rotate her starting line-up, and every outfield player has featured at some point in this tournament. “People get hung up on starters and the 11 and that number rings through a lot of media’s head, but as a coach I look at 14 or 15 starters,” Ellis said when asked about starting Mewis over Horan. “That’s the beauty of having such a deep roster.”

The trouble the Americans had against Spain will surely rub some of the shine off their impressive performances in the World Cup so far. It is perhaps lucky for the Americans that the side they face next, France, were hardly convincing winners against Brazil on Sunday.

Still, USA v France figures to be the marquee match of the quarter-finals. “This is the game everyone had circled,” Rapinoe said. “I hope it’s wild and crazy and the fans are crazy and there’s tons of media around it and it’s a big spectacle.”