Tournament debutant Anderson leads the group with two wins and would have been assured of a semi-final spot had Federer been beaten by Thiem or even dropped a set.
The heavy loss came less than 48 hours after world No. 9 Nishikori handed 20-time Grand Slam champion Federer his first straight-sets loss in round-robin play at the tournament.
Nishikori was staring down the barrel of the ultimate humiliation, and only the second 6-0 6-0 defeat in the history of the Finals, following a ruthless display by the South African.
If Dominic Thiem beats Roger Federer tonight or Roger prevails in three sets, Kevin will be the first player to secure the place in the semis.
The 32-year-old debutant at the ATP's blue riband event won the first 11 games of the round-robin clash, before a relieved Nishikori finally got on the scoreboard with an angled victor. "I'm happy with how I played, and it was good fun against Dominic", Nitto ATP Finals quoted Federer, as saying.
With Nishikori remaining out of sorts with another 10 unforced errors in the second set, Anderson powered ahead to make it 11 games in a row.
The 28-year-old leads their head-to-head 3-1 and thumped him 6-3 6-1 in Vienna just a couple of weeks ago.
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Thiem looked nervous against the 20-time Grand Slam victor and handed his opponent a break in the first game of the second set.
If Federer looked bad-tempered and off his game in losing to Kai Nishikori on Sunday, Tuesday saw the return of the calm and controlled Federer, who was totally in control of his game.
"Amongst the best I've played", Anderson told reporters when asked to rate his 64-minute masterclass.
"I think I made it really easy today". I don't know. But, yeah, I just didn't play well this week.
"I don't feel like I need to comment on this". I just see somebody across the net who's feeling it right now.
It's also no exaggeration to say that if Anderson performs as he did against Nishikori on Tuesday, then he can beat Federer in this form.
It only delayed the inevitable, though, as Anderson duly served out for a 6-0 6-1 victory in 63 hopelessly one-sided minutes.
Anderson is in pole position, but qualification for the semi-finals is still possible for all four players.
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