Seeded 28th and ranked 147, Myneni had his moments before losing 6-7(5), 5-7 in a second round qualifying match.
Trailing 2-5 in the first set, he staged a splendid recovery, but wasted a set point in the 12th game.
The 188-ranked American not only saved the set point but also erased a 2-5 deficit in the tie-breaker, winning five straight points.
The 28-year-old Indian, who overcame Korea's Duckhee Lee 1-6, 7-6(3), 6-4 in the first round, conceded a 4-1 lead in the second set, but kept fighting and drew breast at 4-4, breaking Tiafoe in the eighth after staving off a break-point.
Eventually, he dropped serve in the 12th game, allowing Tiafoe advance to the second round.
Ramkumar Ramanathan was knocked out on Monday in the first round, beaten 3-6, 3-6 by Matthew Barton of Australia.
Image: Saketh Myneni
Photograph: Twitter
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