Forward Luis Diaz (25) cancelled out Wilfried Zaha's (29) opener for Palace and Nunez was red carded for a head-butt on his home debut.
"I told the boys after the game I'm proud," Klopp told reporters.
"I thought the start (of the second half) was positive again and then the red card happened. But obviously the best possible reaction from (Diaz) then -- worldie, wonderful goal.
"And from that moment on it was a special game," he said, adding that he was pleased with how much pressure they put Palace under with 10 men.
Nunez was shown a straight red card in the 57th minute for his reaction to a shove from Palace defender Joachim Andersen (26), and Klopp said the Uruguayan international needed to learn from his mistake, which will likely result in a three-match ban.
"A little provocation here and there - and definitely the wrong reaction. A clear red card, I cannot deny that. He will learn from that," Klopp said.
"Unfortunately, he has now a few games' time to do so.
"Andersen wanted that, I would say, and he got it. But he made a mistake, Darwin, so of course, we will talk about it."
Liverpool, who are 12th in the table after drawing their first two games, next travel to Old Trafford to face bottom side Manchester United.
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