Mikhail Kukushkin gave the Kazakhs the early momentum with a straight sets 6-4, 6-3, 6-3 win over teenager Thanasi Kokkinakis in just over two hours.
Gasquet, who is ranked 10th in the world, won in a tight three-setter 4-6, 6-4, 6-4 in two hour 23 minutes to record his second victory over Kukushkin in as many meetings.
Jan Hajek of Davis Cup holders the Czech Republic lifted his team 1-0 up against the hosts Kazakhstan in Astana on Friday, beating local favourite Mikhail Kukushkin in straight sets on Friday.
Japan's Kei Nishikori earned his first ever win at Wimbledon as the 19th seed marked his return from injury with a 7-5, 6-3, 6-4 victory over Mikhail Kukushkin of Kazakhstan in the first round on Tuesday.
Second seed Rafael Nadal continued the quest for an eighth straight title at the Monte Carlo Masters as he dispatched Mikhail Kukushkin of Kazakhstan 6-1, 6-1 in exactly one hour.
World number 92 Mikhail Kukushkin wrote his name in Kazakh history with the country's best ever male performance at a grand slam Saturday, thanks to a bizarre victory over injury-hit Gael Monfils.
Top seed Andy Murray survived a tense examination from unheralded Mikhail Kukushkin to scrape into the second round of the Brisbane International on Tuesday.
Japan's Kei Nishikori reached his third semi-final in six weeks with a marathon 6-4, 5-7, 6-4 win over Mikhail Kukushkin at the Swiss Indoors here on Friday.
Fourth-seeded Marin Cilic of Croatia advanced to the second round of the St. Petersburg Open on Tuesday, while defending champion Mikhail Kukushkin of Kazakhstan made an early exit.