Twenty-year-old Tiafoe is probably the most exciting of the crop of young American men coming through and he reached the third round of a grand slam for the second time with a 4-6 6-4 6-4 7-5 victory over Anderson.
The fifth seed, who was beaten in the opening round by Kyle Edmund 12 months ago, did not appear to be feeling 100 per cent and called the trainer on court during the third set.
World No 39 Tiafoe, who is into the third round at Melbourne Park for the first time, equalling his best ever Grand Slam result, next plays veteran Italian Andreas Seppi.
“It means the world to me,” said the American, who will celebrate his 21st birthday on Sunday and is fresh from playing the mixed teams Hopman Cup in Perth with Serena Williams.
“I lost to him three times last year and when I was down a set and a break it looked like it would happen again. I dug deep and went to a different place.
“It’s all about competing, how bad you want it and I want it real bad. I’d love to get to the second week of a Slam for the first time,” he added.
Sixth seed and last year’s Australian Open runner-up Marin Cilic ground out a tough 7-5 6-7 (9-11) 6-4 6-4 win over American Mackenzie McDonald to move into the third round.
“Every single time the atmosphere here is amazing,” said Cilic, who has been struggling with a knee injury. “Mackenzie played unbelievable tennis today, he was hitting from the back ridiculously well. I had to work hard to win this one and I’m extremely pleased.”
Meanwhile, Greek star Stefanos Tsitsipas had a brief hiccup against Serbian qualifier Viktor Troicki. He dropped the second set but moved through to the third round with a 6-3 2-6 6-2 7-5 victory.
Also, aside from Anderson, Gael Monfils was another casualty of Australian Open 2019, losing to Taylor Fritz with a score of 3-6, 7-6, 6-7, 6-7.