The Uruguayan has been out of favour so far this season, but his late brace against the Magpies showcased the best of his qualities and he will get more chances, Klopp told Sky Sports.
"He was obviously fired up about not playing, not starting, and he put all the energy into those two shots ... it was a super-mature performance and two top-class goals," a delighted Klopp said as he reflected on what he described as one of the best victories of his career.
Liverpool went behind midway through the first half and then had captain Virgil van Dijk sent off for a foul on Alexander Isak, and when forward Luis Diaz was replaced by centre back Joe Gomez it looked like Liverpool were battening down the hatches.
However, Klopp decided to have one last throw of the dice, and Nunez responded with two late goals to snatch victory from the jaws of defeat for the Reds, boosting his chances of more playing time.
"It's early in the season, you need to find stability. We need to find a formation you need to get results, and then we play every three days and everybody will play, there's no doubt about that," Klopp explained.
"Definitely, a striker scores two goals and you can see he gets a smile on his face, and rightly so. But for the team, it was super-important that they got something like that, you need these kinds of things to grow together," Klopp added.
Back-to-back single-goal defeats to Manchester City and Liverpool will leave Newcastle playing catch-up in the league, and manager Eddie Howe said it will be a challenge to pick his players up again.
"There was a lot of good, we should have put the game to bed and that is probably the thing we are all kicking ourselves with," Howe told Sky Sports.
"There will be a lot of emotion. For us, we've had three really tough opening games, won one and lost two. The two we lost were lost by the narrowest of margins, and they were against the best two teams in the Premier League," he added.
