Abby Wambach scored the winner for the 1991 and defending champions in Foxboro, Massachusetts to book a semi-final meeting with either Russia or Germany.
Sweden also progressed to the last four on Wednesday, after their 2-1 win over Brazil.
The European side will face either China or Canada for a place in the final on October 12.
U.S. coach April Heinrichs recalled Mia Hamm and captain Julie Foudy after resting the pair for their last group game on Sunday, but it was two younger, less familiar names who ended the hopes of 1995 world champions Norway.
Cat Reddick -- the defender celebrating her 21st birthday on Thursday -- floated a free-kick in to the area after 24 minutes and Wambach steered her header past Norway keeper Bente Nordby for the U.S.'s 10th goal from a set play at this year's tournament.
Hamm could have made the game safe after 67 minutes, but her poor penalty was saved by Nordby.
"We knew along that if we want to win this tournament, we would have to go through some of the best teams in the world," said Heinrichs.
"Now we need some rest before (the semi-final on) Sunday."
PHYSICAL APPROACH
Brazil had looked impressive in the group phase and, in Katia, possessed the tournament's joint leading scorer.
But their love of the short passing game was overwhelmed by Sweden's physical and direct approach.
Victoria Svensson gave the Swedes the lead after 23 minutes with a header, before the South Americans equalised with a 44th minute penalty from Marta, after the 17-year-old had been brought down by Sweden's stand-in keeper Sofia Lundgren.
Malin Andersson secured the victory for Sweden with a 25 metre free kick after 53 minutes.
Brazil thought they had won a last-gasp penalty after Sara Call appeared to trip Katia in the area but their pleas were ignored.
"It was a very physical and difficult game for us, and for the young players on our team, some of them you see were even crying on the field," Brazil coach Paulo Goncalves said.
Sweden coach Marika Domanski-Lyfors said: "I think we played both tactically and technically well, like we should play against Brazil."
In the remaining quarter-finals on Thursday, Russia take on Germany and China meet Canada, with both games in Portland, Oregon.
The semi-finals will be played on Sunday, also in Portland.
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