Roared on by a vociferous Australian contingent on Court Two, the 19-year-old wildcard showed incredible resilience to complete the biggest win of his career.
Having fended off a succession of match points, one of which was saved after a successful Hawkeye appeal on a second serve, he broke former semi-finalist Gasquet's serve at 8-8 in a deciding set that captivated the crowd for 71 minutes.
Serving for the match, he then showed no nerves as he held to love, sealing victory with his 21st ace and flinging his racket to the court in apparent disbelief.
The highly-rated Kyrgios, one of a batch of young Australians hoping to revive the fortunes of the former powerhouse nation, was serenaded by the yellow-clad Aussie fans who were set for a double-header on the bowl-like court with old favourite Lleyton Hewitt in action later on day four.
Kyrgios was one of five Australians to reach the second round of the men's singles - the most since 1999 - and will fancy his chances of further progress when he meets unseeded Czech youngster Jiri Vesely in the next round.
(Reporting by Martyn Herman; Editing by David Goodman and Pritha Sarkar)
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