Wieber had been the favourite for the individual crown coming into the Games after taking the world gold in Tokyo last year, but will now have to content herself with trying to win a team medal and the floor title.
Though Wieber finished fourth on Sunday in the all-around standings, led by Russian Victoria Komova, Americans Aly Raisman and Gabby Douglas were above her and each country is allowed no more than two gymnasts in Thursday's final.
The U.S. team finished a long day of qualifying 1.434 points ahead of Russia, mirroring the men's team placings on Saturday.
China's women were a further 3.792 behind and Romania a close fourth. The scores, though, will be reset to zero for Tuesday's team showdown.
The Chinese team, including He Kexin and Deng Linlin, who had helped to win the team gold at the Beijing Games, looked below par.
Beijing gold medallist He finished behind Briton Beth Tweddle on the asymmetric bars and the team failed to get a gymnast into the top eight of the vault and floor, shutting them out of both apparatus finals.
Yao Jinnan, the world all-around bronze medallist, banged her head and knees in a bad landing from the vault and was bandaged up by her coach.
World beam champion Sui Lu, though, saved some face for China by qualifying top on her favourite apparatus for the final on August 7.
Tweddle, fired up by home advantage and a cheering crowd of soldiers filling previously empty prime seats, got the day's only mark over 16 to lead the qualifiers in the asymmetric bars, the event in which she owns two world titles.
At 27, and competing in her final Olympics, she led Britain to fifth place, with Japan, Italy and Canada making up the final eight who will fight for the medals.
"We got a standing ovation as we walked back along. It was amazing," said Tweddle, who has come back after knee surgery three months ago.
Wieber, 17, paid the price for a series of wobbles on the beam and a slight overstep on the floor as Raisman and Douglas, with whom she won the world team title last year, produced slicker routines.
The 16-year-old Douglas, who beat Wieber in the U.S. trials, had a big overstep on the floor herself, incurring a 0.3-point penalty, but did well enough on the other apparatus for the mistake not to matter.
"Today (Jordyn) wasn't quite as sharp," U.S. team co-ordinator Marta Karolyi told reporters. "Aly's success today just proves that hard work pays off."
"It was always going to be close between the three of us doing all-round and in the end it is what it is," Wieber told the Olympic news service after avoiding the world's media.
"It was hard because of course I wanted that spot but I also wanted Aly to do her best for the team."
Helped by China's failings, however, Wieber qualified in sixth place for the floor final.
Raisman was top of the floor qualifiers while her fellow American and world champion McKayla Maroney led the way into a vault final which will feature 37-year-old Oksana Chusovitina, competing in a record-equalling sixth Olympics.
Komova, who lost out to Wieber for the world all-round title by 0.033 points last October, qualified second for the beam final, behind Sui, and also made the asymmetric bars final.
Romania's Catalina Ponor, a triple Olympic champion who has returned to competition at the age of 24 after an eight-year break from the Games, cried after realising she had reached the beam and floor finals.
Uzbekistan's only representative in the gymnastics, Luiza Galiulina, who had been due to compete on Sunday, was provisionally banned from the Games after testing positive for a diuretic.
Her coach is Uzbek-born Chusovitina, who now competes for Germany, but the oldest woman in the competition declined to talk about Galiulina on Sunday.
Asked, instead, how she felt after getting another crack at a vault medal following her silver in Beijing, Chusovitina told reporters: "Very good, as if I was 18 again. I have got in the final and I'm happy. My age doesn't matter because when you are in the arena everybody's judged the same."
The Olympics of Egyptian gymnast Sherine Ahmed Elzeiny came to a tearful end when she collapsed, clasping her right hamstring, at the end of her floor routine and had to be carried off by her coach.