Djokovic reached the Grand Slam semi-finals in New York for the 12th time after his fourth straight win over Berrettini.
Berrettini only played well in the first set, before losing because of a lack of sudden play.
Just like the Wimbledon final, Berrettini won the opening set and then played gradually worse in the following sets.
In sets two and three, Berrettini could not take advantage of any break points and lost two service games.
Djokovic is two wins away from the record of 21 Grand Slams.
In the fourth set, Djokovic did not let his juniors have any break points and won the first game of blocking the ball.
Zverev extended his winning streak from the Tokyo Olympics to 16 matches, defeating Lloyd Harris 7-6 (6), 6-3, 6-4 in the previous quarterfinal match.
Zverev reached the Grand Slam semi-finals for the fourth time.
Serving was the difference between Zverev and Lloyd Harris in the quarterfinals.
`Serving is the skill I spend the most time practicing,` Zverev said about the main weapon in the game.
The South African tennis player won the ball game twice, in the first and third sets.
Zverev 3-0 Lloyd Harris
In the next two sets, Zverev won one more break point than Harris and maintained the advantage until the end of each set to win 6-3, 6-4.
`Novak is the best player in the world, very difficult to beat,` Zverev said about Djokovic, his next opponent at the US Open.