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The Americans had been 2-0 up after singles wins on Friday, but Williams succumbed to a 7-6 3-6 7-6 loss to Anastasija Sevastova, while Kenin was beaten by Jelena Ostapenko.
But victory for doubles duo Kenin and Bethanie Mattek-Sands in the deciding rubber secured the Americans a spot in April's Finals in Budapest.
Australian Open champion Kenin rarely looked comfortable against a fired-up Ostapenko, and despite a brief revival in the second set, it was the Latvian who took it relatively comfortably in the third set, 6-3 2-6 6-2.
Williams was the favourite against Sevastova in her second singles matches, but hit 30 unforced errors in a shambolic first set before being edged out in a dramatic third set tie-break.
The USA had their doubles pairing to thank as Kenin and Mattek-Sands made short work of the deciding match, winning 6-4 6-0 in an encounter that lasted little more than an hour.
America will be joined at the Finals by Spain, Russia, Germany, Belarus, Switzerland, Belgium, Slovakia, Australia, France, Czech Republic and hosts Hungary.