Scott, 16, from Cape Girardeau, finished in 4 minutes, 39.44 seconds, while Katarzyna Pawlik of Poland won the event in a world-record time of 4:33.15.
"I've been training so hard for this, and it feels great to be on the podium," Scott said in a release from the U.S. Olympic Committee. "I tried to go out as fast as I could and then hold it. It was my first medal ever, and I'm really happy with it."
Scott had posted personal-best records in all five of her previous events despite not finishing in the top three. She was fourth in both the 100 backstroke and the 100 freestyle.
Scott entered the final of the 400 free as the top seed after posting a time of 4:46.16 in the preliminaries earlier Monday.
"It was a good morning swim," Scott said. "I've dropped time in every race I've swum, so I am very happy about that."
Scott, who competed at Central High School during her freshman year and qualified for the state meet, had spent the year leading up to the Paralympics at the U.S. Olympic Training Center in Colorado Springs, Colo. She was born with a form of spina bifida and began competing in international competitions for physically disabled athletes just three years ago.
Jill Kennedy, the Zalma native and Southeast Missouri State graduate competing in her third Paralympic Games, completed her competition Monday without winning a medal.
She finished eighth in the dwarf-only division of the women's shot put after a sixth-place finish Saturday in the discus.
Kennedy, who competed in weightlifting in 2000 in Sydney and in the field events in 2004 in Athens, said this year's event was memorable, including the best of the three Opening Ceremonies and the most energized crowds she has seen.
"It was a disappointment, of course, not to get a medal but not a shock," Kennedy wrote in an e-mail after the discus competition. "I threw OK, but was trying to be too perfect.
"The top four finishers all had very impressive marks, so there's no shame in losing to them."