Six-time champion Roger Federer beat Kevin Anderson 6-4 6-3 to secure his place in the semi-finals of the ATP Finals for the 15th time.
Federer’s victory means he tops his group, boosting his chances of avoiding world number one Novak Djokovic in the last four in London.
The Swiss, 37, looked out of sorts when he lost his first match on Sunday but bounced back with two successive wins.
South African Anderson had already won two matches to reach the semi-finals.
Top seed Djokovic is already through to the semi-finals and can clinch top spot in his group by beating Marin Cilic on Friday.
Alexander Zverev, John Isner and Cilic are battling for the other semi-final berth.
Meanwhile, Britain’s Jamie Murray and Brazilian partner Bruno Soares completed a clean sweep of wins in their doubles group by beating Henri Kontinen and John Peers 3-6 7-6 (7-3) 10-3.
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Questions were asked about Federer’s form and state of mind after he lost to Kei Nishikori in straight sets on Sunday but he gave himself a chance of reaching the semi-finals by beating Dominic Thiem on Tuesday.
And after Thiem defeated Nishikori earlier on Thursday, the Swiss knew he needed to win only six games to join Wimbledon finalist Anderson in the semi-finals.
“I’m very happy, my first match was tough and I never got going, but with my back against the wall maybe it’s easier for me to play,” said Federer.
“Kevin’s had a great year, but it’s been a fun group and I’m very happy to be in the semis.”
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Federer had three break points on Anderson’s formidable serve in the seventh game of the first set and secured the break when the world number six double-faulted.
Surprisingly Federer shanked two shots to lose his serve in the next game but the Swiss broke again to love as Anderson temporarily lost his range.
Federer had to save three break points at 5-4 but held his serve to take the set – much to the delight of the majority of the fans inside the O2 Arena.
The 20-time Grand Slam winner looked back to near his best in the second set, playing nearer the baseline and moving his opponent around.
And Federer went on to get a small measure of revenge for his Wimbledon quarter-final defeat at the hands of Anderson by wrapping up victory in one hour and 16 minutes.
“These round-robin formats are not straightforward, we’re used to it being you lose, you leave, you don’t hang around,” added Federer, who is aiming to win the 100th title of his career.
“Maybe it was difficult for Kevin having already qualified, whereas with Thiem winning it was maybe easier for me.
“But I’m happy I’m still alive and hope I can play a good match the day after tomorrow.”