The U.S., who finished as runners-up in the 4x400m two years ago after four successive world titles, were anchored by Natasha Hastings and finished 20 metres ahead of the British team, in a world leading time of 3 minutes and 21.66 seconds.
Britain's quartet qualified in 3:24.74, with Botswana clinching the third automatic place in the first heat to make it through to Sunday's final.
There was heartbreak in the second heat for the Bahamas, running without Olympic 400m champion Shaunae Miller Uibo, as Christine Amertil stumbled and fell 200 metres into her second leg. The team did not finish the race.
The heat was won by Jamaica in 3:23.64, with Nigeria and Germany coming second and third.
Poland and France also qualified for the final as the fastest losers.
(Reporting by Christian Radnedge; Editing by Toby Davis)
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