Musovic kept the Scandinavians in the match, making 11 saves before the shootout to deny the Americans and ensure the game ended 0-0 after extra time.
"I'm at a loss for words. I don't know what to say except that I am extremely proud that we persevered against an extremely strong U.S. that pressed for 120 minutes," Musovic told reporters. "We're not going home yet."
Sweden captain Magdalena Eriksson praised the Chelsea keeper.
"Zecira had a great game, we hung on, and we were in the game, and then anything can happen," she said in a radio interview.
"I think it shows the character in the girls today, our play wasn't working but we didn't give up, we battled on, we defended our area very well."
Lina Hurtig struck the winning spot-kick, though U.S. goalkeeper Alyssa Naeher parried the ball on the line and the result needed to be confirmed by VAR.
"She saves it but I get a hell of a backspin on it and don't see if it goes in but my first reaction is that it's in," Hurtig told Swedish newspaper Aftonbladet.
The Swedish players admitted they endured a nervous wait.
"I told Sofia (Jakobsson) that I couldn't bear to watch. I had my forehead on her shoulder," forward Fridolina Rolfo said.
"But she said I had to watch. And then it becomes so special, with an eternal wait for the announcement."
Sweden play Japan in the quarter-finals on Friday.
