The Reds brought striker Nunez to Anfield from Benfica last month for a transfer fee which could rise to a club-record 100 million euros (£85 million, $105 million).
The Uruguayan, who scored 34 goals last season, had been criticised by some supporters after failing to score in Liverpool’s first pre-season games against Manchester United and Crystal Palace.
But he came on at half-time against Leipzig in Germany with Liverpool leading 1-0 thanks to a Mohamed Salah goal and scored a penalty before going on to net another three times.
“Mo gives Darwin the penalty and Darwin, Pandora’s box was open,” Klopp told liverpoolfc.com. “That’s of course a perfect night for him.
“We always think that if you pay a lot of money then the players feel no pressure or whatever.
“They are all completely normal human beings and the first touch is not perfect then all of a sudden… This generation of players read social media, which is really not smart, but they do.
“All of a sudden you get in a rush and these kinds of things. That’s the best way obviously to stop all these discussions.”
Liverpool, who won the FA Cup and League Cup titles last season but finished as runners-up in the Premier League and Champions League, face Manchester City in the Community Shield curtain-raiser next weekend.
The 23-year-old Nunez will be expecting to be a key member of Klopp’s squad after the departure of Sadio Mane to Bayern Munich.
“I think everybody is really happy for him and was looking for him (in the dressing room),” centre-back Joel Matip said.
“Great finish, great speed and the body he always puts in –- I think it will be really difficult for our opponents.”
Liverpool open their Premier League campaign at promoted Fulham on August 6.