The Americans are a win away from April's semifinals after racing to a 2-0 advantage over nervous hosts France while defending champions Italy were level at 1-1 with World Group debutants Ukraine in their best of five encounter.
Russia's Svetlana Kuznetsova swatted aside Ivanovic 6-1, 6-4 but Serbia drew level thanks to Jelena Jankovic.
Czech Republic and Germany had also won one match each in their tie ahead of Sunday's reverse singles and doubles. American Bethanie Mattek-Sands gave last year's runners-up the lead when she edged out an anxious Alize Cornet 7-6 (9-7), 7-5.
The US, again without the Williams sisters, went further ahead with Melanie Oudin's 6-4, 6-4 win over Pauline Parmentier. Former world number one Ivanovic, who lost in the second round at last month's Australian Open, has tumbled down to 23 in the rankings and was never in the match against Kuznetsova.
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