The Greek sixth seed, 17 years younger than his 38-year-old opponent, beat the Swiss at the Australian Open but had fallen to him twice since then.
Federer was unrecognisable from the player who dominated Novak Djokovic in his final round-robin match, struggling on serve and hitting a total of 26 unforced errors compared with just five against the Serbian.
"I really enjoyed myself on the court and sometimes in matches like these you wonder how you recover from difficulties and break point down.
"It is a mental struggle and I'm proud how many I saved today, I was trying not to give an easy time to Roger. He was playing well."
Federer was in deep trouble when Tsitsipas broke him to love in the third game of the second set but he finally made a break point count - his 10th, to level at 2-2.
At 5-4 down Federer knew he had to break Tsitsipas for only the second time in the match.
"I'm frustrated I couldn't play better, and when I did and fought my way back, I threw it away again," said the Swiss.
A year ago, Tsitsipas won the Next Gen ATP Finals. Now, on his tournament debut he is one match away from winning the season-ending event, featuring the year's best eight players.Read more: CNA
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