Hingis, the 32-year-old former top-ranked singled player whose last tour match was September 2007, consistently showed the deft hands that made her a nine-time Grand Slam doubles champion before a sparse crowd that gave her a warm ovation when she was introduced.
Hingis didn't touch the ball until the fifth point of the match but flashed her doubles talent when she hit a backhand volley right at Jurak for a point in the second game.
"I felt like I got more comfortable (after the first set), returning, serving, everything," she said. "For me, mentally, I felt really good after that. The (first) service game of the second set, after that game I was like, 'I'm back to normal.'"
Hingis is scheduled to play doubles with Hantuchova in four tournaments, including the U.S. Open.
"It was a great test for us and being able to beat them the way we did, it just gives us a lot of hope for the coming weeks," Hantuchova said about beating the team that reached the final of the Bank of the West Classic on Sunday.
The 'Swiss Miss' has retired twice previously, the first time in early 2003 at 22 and the last time in November 2007.
The seventh-seeded Ivanovic, from Serbia, won nine of the final 11 games. She used a strong ground game to overpower Cibulkova, the Slovak coming off a victory Sunday in the Bank of the West Classic at Stanford.
"As soon as the draw came out I thought it was one of the toughest first-round matches, especially being a seeded player," said Ivanovic, the former top-ranked singles player and 2008 French Open champ. "But I was really happy with my performance."
Top-seeded Victoria Azarenka of Belarus beat Italy's Francesca Schiavone 6-2, 6-3 in the second round. Azarenka was playing her first match since withdrawing from her second-round match at Wimbledon because of knee and hip injuries.
"Just going on the court and competing was so much joy for me," Azarenka said. "I was really playing and happy to be doing something I was training for. . . . I don't think about it (injuries)."
Ivanovic, who fired coach Nigel Sears after her second-round loss at Wimbledon, said she has been working on "being consistent and also working on that confidence part. It plays a big role. I felt like my game has been there at times but not my confidence."
Vinci beat American Bethanie Mattek-Sands 6-4.