Zverev, 20, has yet to reach the quarterfinals of a Grand Slam, and fell in the first round of the French Open and the second round of Flushing Meadows, having won a Masters title before each major tournament.
Following his five-set defeat to Hyeon Chung, he acknowledged that the pressure affected him and said he would have to work on understanding "what happens to me in deciding moments in Grand Slam," noting, "It happened at Wimbledon. It happened in New York. It happened here."
But Zverev also said he was still learning, adding that Federer had helped by stopping to speak to him in the locker room.
"I just thought some nice words would maybe cheer him up," said Federer. "He looked crushed when I saw him."
Federer was 21 when he reached his first Slam quarterfinal, and then went on to win his first major.
"I know people talk, but for the player, it's not easy if you've never been there. I remember I had a hurdle to [get beyond] the quarters," he said. "That's what I told Sascha... Don't put yourself under unnecessary pressure.
"You're working hard. You're doing the right things. It's maybe not paying off at the Slam level, but just stay calm, don't dig yourself in a hole. There's no reason, for that right now, I think."
The 19-time Grand Slam champion also suggested young players could benefit by concentrating on smaller goals.
"He's still very young, still up-and-coming to some extent. He's still seeking the breakthrough on the Grand Slam level," said Federer. "I just think it's important to sometimes take a step back and actually see the good things you've done, give yourself time, maybe set the bar a bit lower. First let's maybe try to look for a quarters or a semifinals, not just right away think coming to the Australian Open, US Open, I have to win this thing."
Zverev is the youngest player in the Top 10 in the ATP rankings.
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