The USA team finished top of the pile with a score of 275.342, ahead of Russia, Britain, Germany and Japan, with defending Olympic champions China down in sixth place.
Ukraine and France complete the eight countries who will compete in Monday's Team final.
USA athlete Danell Leyva finished first in the Individual All-Around competition, ahead of Russian David Belyavskiy and Germany's Fabian Hambuchen.
Uchimura, a three-time world All-Around champion and double silver medallist from the 2008 Beijing Games, finished ninth in the All-Around qualifications after falling from both the Horizontal Bar and Pommel Horse in an uncharacteristically inconsistent performance.
Despite that, the star of last year's World Championships in Tokyo will compete in the Individual All-Around, Floor and Parallel Bars finals.
Defending Olympic Team champions China also had an off day, finishing second to an impressive British team in the first subdivision of the day before later being pushed down the rankings to sixth.
However, Zou Kai will defend his Olympic Floor title after he finished top on the apparatus with a score of 15.833 while countryman Chen Yibing, reigning Olympic Games Rings champion, qualified for that apparatus with a huge score of 15.858.
Japan led the way on Parallel Bars, claiming first and second in qualifications through Yusuke Tanaka and Kazuhito Tanaka.
On the Horizontal Bar, Holland's Epke Zonderland held on from the first qualification session to finish on top of the pile with a massive 15.966, narrowly ahead of China's Zhang Chenglong and Leyva.
Germany's Marcel Nguyen qualified for the Floor and Parallel bars finals, with Philipp Boy missing out on every final but Monday's Team event after he looked to have picked up an injury on the first rotation.
Elsewhere, Bulgaria's Iordan Iovtchev, competing in his sixth Olympic Games, qualified in eighth and final place in the still Rings competition with a score of 15.308.
The 39-year-old's best result was a silver medal on the same apparatus in Athens 2004 and he faces tough competition to get on the podium during the London Games.
Hungary's double world Pommel Horse champion, Krisztian Berki, competing solely on his specialist piece of apparatus, could only score 15.033 but still made the Individual final.
Berki is one of Briton Louis Smith's biggest rivals for Pommel Horse gold and relies on producing stylish routines rather than completing those with a high start value.
Smith, a bronze medal winner four years ago, topped Pommel Horse qualification with a routine scoring 15.800, but Croatian Filip Ude, who won silver in Beijing, did not qualify for the final after he fell off the apparatus and only scored 13.933.
France's Cyril Tommasone, world Pommel Horse silver medallist, did make the cut with a 15.333 routine.