Anderson's win clinched the tie for South Africa following Lloyd Harris's 6-4, 6-4 victory over Nicolas Jarry.
Raven Klaasen and Ruan Roelofse made it a clean sweep for South Africa when they beat Garin and Jarry in the doubles 1-6, 6-3, 10-7.
Anderson, 33, missed the second half of last year after suffering a knee injury at Wimbledon in July, with the South African undergoing surgery in September.
He played his first match back against world number two Novak Djokovic on Saturday, pushing the 16-time Grand Slam winner to the brink before falling 7-6 (7/5), 7-6 (8/6).
Djokovic noted that Anderson's absence "didn't really affect his game that much".
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And after his 74-minute demolition of world number 33 Garin, there is little doubt that the towering South African will be a dangerous unseeded floater in the Australian Open draw.
Anderson, who stands at 2.03 metres fired down 11 aces in his win over Garin and did not face a break point.
"My best ranking is top five, so I would like to get back there," he said.
"But the two biggest goals is I want to win a Masters series. I feel like that's something that I'm definitely a good enough tennis player to do that. I've put myself in good positions but haven't quite taken that step.
"And then, obviously, the grand prize in tennis, a Grand Slam. I've been in two finals. That's the ultimate goal for me. It's one that I really feel like I have a good chance of doing it."
The 22-year-old Harris, coming off his best year in 2019, was impressive in his win over a player ranked 22 places above him.
"I wanted to keep going for my spots on my serve, try to stay as aggressive as possible on the forehand, but the idea was to make him play that extra ball, getting in good defensive positions when he was playing with very strong ground strokes and I think I defended really well," Harris said.
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