Both players traded breaks in the first set. Thompson held serve to lead 6-5 and then broke the Indian to secure it 7-5.
In the second, Bhambri seized control early and stamped his authority by winning it at 6-3.
In the decider, the Australian, world no.103, showcased his skills, hitting with power and precision to jump to a 3-0 lead.
Bhambri rallied to win the next three games to restore parity.
The Aussie secured a break in the 7th game only to be broken right back.
Bhambri dropped serve in the 11th game and the top-seed held his nerve to close out the over two-and-half hour match.
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