Federer was leading four points to three when Zverev stopped the rally after a ball boy had dropped a ball at the back of the court behind Federer.
Zverev, who had taken the first set 7-5, won the replayed point with his eighth ace of the match and surged on to win the shootout seven points to five to advance to his first final in his second appearance at the eight man end of season championships.
“Alex apologised to me at the net,” said Federer. “I was like: ‘Buddy, shut up. You don't need to apologise to me here. Congratulations on a great match and a great tournament so far. All the best for the final.’”
Zverev, 21, will be the youngest finalist at the ATP Finals since Juan Martin del Potro in 2009. He will play either the top seed Novak Djokovic or the fourth seed Kevin Anderson.
Sections of the 17,000 crowd at the 02 Arena in south-east London jeered Zverev during his on court post-match interview with Annabel Croft. Choking back tears, he apologised for the end to the match before Croft berated spectators for not knowing the rules and a lack of respect.
“He shouldn't be apologising,” said Federer reflecting on the match. “He didn't do anything wrong. He just called it how it was. He felt it affected play. There is a rule that if something like this happens, obviously you replay points.
“You don't like to hear booing," added Federer. "But, look, it happens. I'm sure he won't be booed coming out for the final, or at least I hope not. It's not his mistake.”
Federer had been attempting to win the tournament for a record seventh time and with it his 100th title.
The century will have to wait until next season, said the 37-year-old who declared himself happy with a season in which he collected his 20th Grand Slam tournament, six more than his idol Pete Sampras
“Sampras once upon a time said: ‘If you win a slam, it's a good season.’ So I started great. I played super well in Australia again. I have high hopes to always do well. So I'm happy I gave myself opportunities again in that second half of the season.
“I maybe lost a couple of close matches that could have changed things around for me a little bit.
“I’ve just lost in London in the semi-final and so I'm a little bit disappointed because I believe I was close. But being close makes me believe I can keep going and that I can win again. That's uplifting in some ways.”