Basilashvili, who was the only seeded player in action on the opening day, dropped serve once in the first set before finding his feet and converting four of eight break points.
Taylor Fritz was pushed the distance by Spain's Fernando Verdasco, with the American coming out on top in two hours and 40 minutes with a 7-6(2) 6-7(5) 6-4 victory.
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Fritz looked set to win the match in the second set before the 35-year-old Spaniard fought back in the tiebreak to take it into a decider where he eventually ran out of steam.
Portuguese Joao Sousa was also involved in a three-setter, dropping the first set before beating Serbia's Filip Krajinovic 2-6 6-4 7-6(5).
Local favourite Zhizhen Zhang was knocked out in straight sets by Poland's Hubert Hurkacz who prevailed 7-6(5) 6-4.
The top eight seeds - including Novak Djokovic, Roger Federer, Daniil Medvedev and Dominic Thiem - are not in action until the second round after receiving byes.
Djokovic and Thiem were in action in different tournaments on Sunday, winning the Japan Open and China Open respectively.
Former world number one Andy Murray, who was awarded a wildcard last month, plays Argentina's Juan Ignacio Londero on Monday while Rafa Nadal has skipped the tournament with a wrist injury that also cut short his Laver Cup campaign last month.
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