Wimbledon finalist Anderson took a medical time-out mid-way through the four-set match before becoming the highest ranked player to be ousted from the first major of the year.
His young American conqueror, ranked 39 in the world, saved two break points while closing out the match.
"I knew he was going to make me play a tough game," Tiafoe told reporters.
"He returns well. He returns hard."
After sealing match point against the South African, Tiafoe slapped his right bicep, thumped his chest and roared at the terraces.
The victory confirms the 20-year-old as a member of tennis' new generation of men's players who can challenge the game's long-dominant players, including Roger Federer, Novak Djokovic and Rafa Nadal.
"These are the matches where they kind of define you and help you feel more and more comfortable to keep winning matches like that," Tiafoe said.
Formerly an outstanding junior, Tiafoe took Federer to five sets in the first round of the 2017 U.S. Open.
He will play Italy's Andreas Seppi in the third round, which will at least equal his best performance at a Grand Slam, having made the third round of Wimbledon last year.
(Reporting by Jonathan Barrett and Ian Ransom in Melbourne; editing by Amlan Chakraborty)
