The two sides laboured in the first half, with Emma Hayes's team enjoying plenty of possession but struggling to create any meaningful chances at Parc des Princes.

United States improved after the break and its first decent opportunity arrived in the 63rd minute when Mallory Swanson exchanged a one-two before curling an effort just wide.

Rodman's influence grew as the game wore on, but it was Sophia Smith who had the best chance prior to the winner, only for the 23-year-old to take too many touches when through on goal, allowing Ayaka Yamashita to close down the angle.

But as penalties approached, Rodman stepped up in the second minute of stoppage-time at the end of the first half of the additional 30 minutes, collecting Crystal Dunn’s pass before cutting inside and curling in an unstoppable shot.

Japan substitute Riko Ueki almost equalised from close range, but United States was able to clear its lines and hold firm to tee up a semi-final against either Canada or Germany.

Despite neither side creating much in the way of chances with United States mustering an expected goals (xG) of 0.68 to Japan's 0.53, the Americans dominated the ball for large swathes of the match.

They had 71.3 per cent possession, had more than double the passes and final third entries of Japan with 1,026 to 403 and 89 to 44 respectively.

But it ultimately took a moment of brilliance from Rodman to settle the contest. At 22 years and 75 days old, she is the youngest player to score an extra-time goal for the USWNT since Heather O'Reilly at the 2004 Athens Olympic Games.