The third-seeded Tursunov, trailing 4-1 and serving 15-40 in the second set, saved his serve, then went on to win six of the last seven games against his fellow Russian doubles partner.
"I wasn't serving well in the second set," Tursunov said. "I don't know what my first-serve percentage was [46 percent]. I don't know if it goes that low.
"I think he [Kunitsyn] tried to do too much with second serves. The key was to not get upset. You try to correct things as you go."
"I wanted to play on Tuesday, but they said that was impossible," said the 25th-ranked Frenchman. "I did not expect to play my best match today. So I just did my best."
Simon and Mahut spent much of their match battling in long baseline volleys. Simon scored points when he went to his backhand.
"The first set was a great set for me. I was feeling good on the court. The second set was really hard. Every shot was hard to play. I was feeling the heat," Simon said.
Simon said he's in the United States to adapt to the heat and humidity and to get accustomed to the courts in preparation for the masters tournaments. He plans to play at Toronto and Cincinnati before heading for the Olympics, then returning to the U.S.
Also on Monday, Yen-Hsun Lu of Taiwan outlasted qualifier Justin O'Neal of the United States, 7-6 (2), 7-6 (12), staving off three set points late in the tiebreaker.
In a battle of qualifiers, Australian Joseph Sirianni won when his opponent, Russia's Mikhail Ledovskikh, withdrew in the second set after rolling an ankle. Sirianni won the first set 6-3, but was trailing 2-1 in the second set.