It was the first loss at home for Kukushkin, who has a 12-6 record in the competition.
The Czech Republic is without Tomas Berdych, who has a shoulder injury. Their No. 2 player, Radek Stepanek, who had a neck operation in January, was rested for the opening singles. He's scheduled to play doubles on Saturday and may step in for a reverse singles match on Sunday.
The Czech Republic beat Spain in the final last year for its first title as an independent nation. Kazakhstan is looking for its first semifinal appearance.
The winner of the best-of-five series will face either Argentina or France in the semifinals in September.
Argentina ties France 1-1
France has beaten Argentina in all five previous Davis Cup showdowns. The Argentines are aiming to win their first Davis Cup title.
Simon was filling in for Richard Gasquet (injured right ankle). But Monaco delivered when he had to, winning the first set in front of noisy home fans at the outdoor clay venue of Parque Roca on the outskirts of the Argentine capital.
Berlocq got back into the early match in the fourth set by converting two of 10 break-point chances. But Tsonga responded by breaking twice to close out the match.
Information from The Associated Press was used in this report.