By Mark Meadows
The Americans are a win away from April's semi-finals 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.
American Bethanie Mattek-Sands gave last year's runners-up the lead when she edged out an anxious Alize Cornet 7-6(7) 7-5.
"I was really nervous when coming on the court. I guess I take things too much to heart. I felt like I was petrified by the stress," France's Cornet told reporters.
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.
Despite the support of the Belgrade crowd, Ivanovic produced 21 unforced errors in the first set alone before world number four Kuznetsova comfortably closed out the second set.
Jankovic leveled matters with a dogged 4-6 6-4 6-0 win over Alisa Kleybanova in the second rubber.
Italy started badly with Alona Bondarenko crushing Francesca Schiavone 6-1 6-4 in exactly an hour and a half.
The Ukrainian world number 26 cruised through the first set on the hardcourt in Kharkiv before 18th-ranked Schiavone fought back to go 3-1 up in the second.
Powerful Bondarenko then increased the tempo and retook the initiative with a series of booming shots, sealing victory when the Italian went wide with a forehand.
Sister Kateryna Bondarenko fared less well against world number 12 Flavia Pennetta in the second match, losing 7-5 6-3.
Pennetta, who struggled with the partisan crowd, often struck the net in an unconvincing victory.
Germany's Anna-Lena Groenefeld confidently beat Czech 23-year-old Lucie Safarova 6-2 6-2 in Brno but the home team leveled the tie at 1-1 thanks to Petra Kvitova's battling 6-4 6-4 win over Andrea Petkovic.
In World Group II, where the winners playoff against the World Group losers in April, Australia were level at 1-1 with Spain, Estonia led Argentina 2-0, Slovakia were 2-0 up against China and Belgium were drawing 1-1 in Poland.
(Additional reporting by Pierre Serisier; Editing by Dave Thompson)