After both teams were locked at 2-2 after the regulation 90 minutes and extra time, Dutch striker Vivianne Miedema and defender Aniek Nouwen had their spot kicks saved by Naeher.
Megan Rapinoe made it four out of four from the spot for the U.S. as the four-time gold medallists set up a semi-final clash with Canada.
Arsenal forward Miedema has been the player of the tournament so far and netted twice for the Dutch to become the first female player to score 10 goals at a single Olympic Games.
The Netherlands had the perfect opportunity to seal victory in the 81st minute when Kelley O’Hara gave away a penalty but Lieke Martens’ shot was saved by Naeher.
Canada needed goalkeeper heroics of their own to reach the semi-finals for the third consecutive Games as Stephanie Labbe stopped two Brazilian spot kicks en route a 4-3 shootout win.
After Canada captain Christine Sinclair’s penalty was saved by Barbara, Labbe rescued her side with two spectacular saves to keep out Brazil’s final two spot kicks from Andressa and Rafaelle.
Labbe, who made four crucial saves in a goalless 120 minutes at Miyagi Stadium, did well to stay on the field after falling hard at the end of the second half of extra time.
The Rosengard goalkeeper’s heroic efforts ensured Brazil were denied another crack at the gold medal, having claimed silver in 2004 and 2008.
Britain suffered heartbreak in extra time as a Sam Kerr double helped Australia to a 4-3 victory at the Ibaraki Kashima Stadium after her first goal in the 89th minute had brought the scores level at 2-2 to force the extra period.
Ellen White netted a hat-trick for Britain to take her goalscoring tally to six for the tournament but her team mate Caroline Weir had her penalty saved by Teagan Micah in extra time, leaving the door open for a late surge from the Matildas.
Mary Fowler’s heavily deflected strike found the net in the 103rd minute to halt Britain’s momentum before Kerr headed in her second goal of the evening to seal victory.
The Matildas will next face Sweden, who maintained their perfect record in the tournament with a dominant 3-1 win over hosts Japan at the Saitama Stadium.
Goals from Magdalena Eriksson, Stina Blackstenius and Kosovare Asllani ensured Sweden’s bid to improve on their Rio 2016 silver medal remained intact.