The German secured the scalp of his career when he beat the World No2 7-5 7-6 in the semi-finals.
But the pulsating match was overshadowed by an incident at 4-3 in the second set tie-break.
The 21-year-old stopped playing during midway through the point when he noticed a ball boy had dropped a ball at the back of the court.
Neither the umpire nor Federer had seen the incident but Zverev took the decision to stop.
The O2 crowd reacted in disgust but because it is in the rules, the umpire agreed with the decision and replayed the point.
Zverev sent down a 137mph ace and went on to clinch the match and a place in tomorrow's final against either Novak Djokovic or Kevin Anderson.
Zverev said: "First I want to apologise for the situation in the tie-break.
"The ball-boy dropped the ball. It's in the rule and we have to replay the point.
"I apologised to Roger at the net and he told me it's okay. He knows it's in the rules.
"There are a lot of Roger fans here and given what he has achieved, he has more fans in the world. I didn't mean to upset anybody. All I can say is sorry.
"It was a tough situation for everybody. I just wanted to say sorry for the situation and we had to replay the point. I aced him afterwards which probably didn’t help his situation but it helped me."
Despite the controversy in the final stages, the game was a brilliant spectacle for those fortunate enough to be in the Arena.
Both players tried to go on the front foot and take it to their opponent.
After no break points throughout the first set, suddenly Zverev found himself 0-40 up on the Federer serve at 5-6.
And when the Swiss legend pulled a routine forehand into the tramlines, he knew he had to go the distance if he was going to win his 100th career title tomorrow.
He fought back and got an early break in the second but immediately lost the advantage.
With Zverev serving excellently, it looked to be going to a breaker.
And when it did, the pair could not be split until Federer sunk a routine volley into the net at 5-4 to gift the German two match points.
A driving backhand volley sealed the win and Zverev was through, to the disappointment of the partisan crowd.
Event director Adam Hogg tweeted his disgust with the spectators to his tournament, some of whom will have forked out £130 to be there.
He said: "Astonishingly disappointing reaction from the crowd. Huge credit to Sascha Zverev for reaching his first final here at the #NittoATPFinals.
"And well done @Annabel_Croft on handling the situation with aplomb."
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