Seeded 28th and ranked 147, Myneni had his moments before losing 6-7 (5) 5-7 in the second round contest that kept oscillating from one extreme to the other.
At one stage, Myneni almost rose from the dead as he was trailing 2-5 in the opening set but stretched it to a tie-breaker before letting go a set point in the 12th game.
Tiafoe, ranked 188, showed tremendous willpower to fight even when he was on backfoot in the tie-breaker as he was staring down the barrel trailing 2-5. He reeled off five straight points to nose ahead.
Eventually, Myneni dropped serve one final time in the 12th game to let Tiafoe advance to the second round.
Ramkumar Ramanathan was knocked out on Monday in the first round itself.
In the opening set, Myneni had got an early opportunity to take lead but squandered three break chances in the second game after an easy hold.
Tiafoe though did not waste opportunity that came his way as he cashed in on the first break chance in the third game to create a mini-opening.
Serving for the set at 5-2, Tiafoe lost his serve and Myneni saved four set points in the gruelling ninth game to narrow the gap to 4-5.
Stepping up the pressure, the Indian broke Tiafoe in the next game to bring the match to even keel at 5-5. An easy hold at love for Myneni in the next meant that the American was not serving to stay in the set.
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