The USA threw down the gauntlet to defending champions China - who qualified third - with a total of 181.863, while Russia took second.
Russian Viktoria Komova topped the All-Around qualification but American Jordyn Wieber missed out due to the two gymnasts per country rule.
Wieber finished behind team-mates Aly Raisman and Gabby Douglas but will compete in the Floor final.
Raisman also topped Floor qualification ahead of Russia's Sandra Raluca Izbasa and Italy's Vanessa Ferrari.
Russia's Aliya Mustafina, who ruptured her anterior cruciate ligament in the 2011 European Championships on the Vault, had a good day as she qualified for the All-Around, Floor and Bars final.
World Beam champion Sui Lu of China finished first on the apparatus with an impressive score of 15.400 ahead of Komova and Douglas, while world Vault champion McKayla Maroney led the way on that apparatus with a 15.800 score.
Britain's Beth Tweddle headed up qualification for the Uneven Bars with a stunning routine of 16.133 ahead of world Bars champion Komova and defending Olympic champion He Kexin.
Japan's Koko Tsurumi was the only member of the team to qualify for an apparatus finals, the Uneven Bars, but Asuka Teramoto and Rie Tanaka qualified for the Individual All-Around.