"I just try to be calm," Rapinoe said after. "I say to myself, the worst that's going to happen is that we lose the whole thing."
Vivianne Miedema, who was making her 100th appearance for Netherlands, saw her penalty saved by Naeher to hand the advantage to the USWNT before Rose Lavelle, Alex Morgan, Christen Press and Rapinoe stepped up to seal victory.
"I felt amazing. I should have taken the penalty in the game," Miedema said. "I should have just stuck with my plan and go from there. But it's life. It's football and I just need to move on."
Miedema, who has been one of the standout players of the tournament, scoring 10 goals in total, had scored in each half for Netherlands. Her tally is the highest ever achieved by a women's player in the Olympics.
The first came after 18 minutes with her second touch of the match. The USWNT were on top, and despite a crowded box she managed to get her shot off and on target.
Williams got on the scoresheet herself two minutes later with a right-footed shot past Sari van Veenendaal.
The USWNT's lead didn't last long into the second half with Miedema scoring her second on 54 minutes. Naeher nearly saved it, but there was too much power on the shot.
Netherlands had the perfect opportunity to put the game to bed on 81 minutes when Kelley O'Hara gave away a penalty, but Lieke Martens' shot was saved by Naeher.
"This team just kept pushing for 90 minutes, 120 minutes, and we just kept believing that we were going to find a way to get it done," Naeher said. "Very proud of the four players to step up and score their four penalties to go four-for-four. That is huge."
Lineth Beerensteyn had the ball in the net four minutes into extra time for Netherlands, but it was ruled offside and chalked off by VAR.
The USWNT had two goals ruled out by VAR in the second half of extra time. Christen Press was judged to be offside on 109 minutes and then Morgan was found to be offside four minutes later had Rapinoe had played her through clear on goal.