The defending champion United States and Czech Republic were one more win away from making the Fed Cup semifinals on Saturday.
The competition's most successful teams won their opening singles in World Group first-round home ties, the Americans against the Netherlands, and the Czechs against Switzerland.
The two other ties, France-Belgium and Belarus-Germany were 1-1.
After Venus Williams won her 1,000th singles to open the U.S. account, Coco Vandeweghe almost blew it in Asheville, North Carolina.
Vandeweghe blew a 4-2 lead in the first set against Richel Hogenkamp, ranked outside the top 100, and went down 3-0 in the second, smashing her racket in the process.
Vandeweghe fought back to serve for the second set at 5-4 but dropped serve, and had to take it in a tiebreaker.
That finally broke Hogenkamp's poise, as the Dutchwoman faded in the third set and Vandeweghe won 4-6, 7-6 (6), 6-3. Despite 56 unforced errors, she won her 13th straight Fed Cup match.
Williams beat Arantxa Rus 6-1, 6-4, her 776th win in her 1,000th singles, 24 years after her first.
"I'm not really going for milestones, but then they happen," Williams said.
She could clinch the tie on Sunday in the first reverse singles, against Hogenkamp. That could make the return of sister Serena Williams, after more than a year away from the game while becoming a mother, irrelevant. Serena is scheduled to play doubles, the last match.
In Prague, Petra Kvitova and Barbora Strycova put the strong Czechs in pole position.
Kvitova defeated Viktorija Golubic 6-2, 1-6, 6-3, and Strycova doubled the advantage against Belinda Bencic 6-2, 6-4.
The Czechs haven't lost to the Swiss in 20 years, but Golubic won both of her singles against the Czechs in 2016 and Kvitova hadn't forgotten.
"I said to myself, I didn't want to be the next (Czech) to lose to her," Kvitova said. She can clinch in the first reverse singles on Sunday against Bencic.
France and Belgium, in their first tie since 2000, were 1-1 without surprise in Mouilleron-le-Captif.
Elise Mertens, ranked 20th, swept aside Pauline Parmentier of France 6-2, 6-1, then Kristina Mladenovic, ranked 13th, beat Kirsten Flipkens of Belgium 6-2, 6-4.
On Sunday, Mladenovic faces Mertens, Parmentier follows against Flipkens, and the French will combine against Belgian duo Ysaline Bonaventure and Alison Van Uytvanck.
In Minsk, the first tie between 2017 finalist Belarus and Germany was at 1-1.
After Aryna Sabalenka beat Tatjana Maria, playing her first tie for Germany in seven years, 4-6, 6-1, 6-2, Belarus would have expected to go 2-0 up.
But Antonia Lottner, making her Fed Cup debut for Germany, upset Aliaksandra Sasnovich, ranked more than 100 spots higher, 7-5, 6-4.
Belarus President Alexander Lukashenko attended the tie.
"When you know the president is in the stands," Sabalenka said, "you want to show him your best tennis."