Federer walked out into a jam-packed O2 Arena knowing any repeat of his insipid performance on Sunday against Japan's Kei Nishikori could have resulted in an ignominious early exit.
But the 37-year-old again shrugged off the passing of time to swagger past the dangerous Anderson 6-4 6-3 and reach the semi-finals for the 15th time in 16 appearances at the event.
Record six-time champion Federer turned on the style to avenge this year's gut-wrenching Wimbledon quarter-final defeat by Anderson in which he squandered a match point.
Federer and Anderson both ended up with two wins in the Lleyton Hewitt group but the Swiss grabbed first place on the head-to-head record - meaning he is likely to avoid world number one Novak Djokovic in Saturday's last four.
Djokovic will top the other group unless he loses to Marin Cilic and John Isner beats Alexander Zverev.
A tournament that has so far lacked thrills could yet have a Sunday showdown between old warriors Federer and Djokovic.
Earlier on Thursday, Dominic Thiem beat Nishikori 6-1 6-4 - a result that meant the Austrian would reach the semi-finals if Federer lost to Anderson and won fewer than six games.
