Lleyton Hewitt beat Jack Sock 7-5, 6-4, and Hewitt and Wolfe beat Sock and American substitute Victoria Duval 7-6 (4), 6-1 in mixed doubles to win the tie 3-0.
Ukraine leads the group with wins over the Americans and Czech Republic.
Other top-10 women on the injured list included defending champion Maria Sharapova and top-seeded Simona Halep at the Brisbane International, where they withdrew without playing because of injuries within an hour of each other.
At the ASB Classic in Auckland, New Zealand, defending champion Venus Williams and 2014 winner Ana Ivanovic were beaten in first-round upsets.
And at the Shenzhen Open in China, two-time Wimbledon champion and No. 6-ranked Petra Kvitova retired during the first set of her first-round match due to an undisclosed illness.
No. 4-ranked Sharapova was scheduled to play fellow Russian Ekaterina Makarova but withdrew hours before the first-round match rather than risk further damage to a sore left forearm.
The five-time major winner said she'd hurt her forearm in practice and had to withdraw as a precaution. Her place in the draw was taken by Margarita Gasparyan, who lost to Makarova.
Halep withdrew a day ahead of her scheduled opening match because of a problem with her left Achilles tendon.
"This injury is very difficult," Halep said. "I thought I was ready, but I can't do a full match."
Halep said she hoped to play at the Sydney International next week. Sharapova is heading directly to Melbourne to prepare for the Australian Open, which starts Jan. 18.
Sharapova won the Brisbane and Rome titles in 2015 and lost the final of the Australian Open to Serena Williams, who added the French Open and Wimbledon titles and was two match victories from a season Grand Slam when she lost to Roberta Vinci in the U.S. Open semifinals.
Vinci moved into the Brisbane International quarterfinals with a 6-1, 6-1 victory over 2014 Australian Open finalist Dominika Cibulkova on Tuesday. She'll meet either 2009 champion Azarenka or Ysaline Bonaventure in the last eight.
Also advancing were No. 6-seeded Carla Suarez Navarro, who beat Sam Stosur 6-1, 6-7 (3), 6-4, and Anastasia Pavlyuchenkova, who held off Alize Cornet 6-2, 1-6, 6-4.
Among the men progressing in Brisbane were No. 6 David Goffin, who beat Thomaz Bellucci 6-4, 6-4, and No. 8 Dominic Thiem, who held off local James Duckworth 3-6, 6-4, 6-3.
In New Zealand, top-seeded Venus Williams lost to 18-year-old Russian Daria Kasatkina 6-7 (4), 6-3, 6-2, and Ivanovic was beaten by British qualifier Naomi Broady 7-5, 6-4. Of the three former No. 1s in action, only third-seeded Caroline Wozniacki advanced, 6-4, 6-4 over Danka Kovnic.