Klopp's men shone despite the absence of Mohamed Salah and Sadio Mane at the Africa Cup of Nations for the first half hour as they raced into a 2-0 lead thanks to goals from Virgil van Dijk and Alex Oxlade-Chamberlain
However, they needed goalkeeper Alisson Becker to be in top form and a controversial late penalty, converted by Fabinho, to secure a win that closes the gap on Manchester City at the top to nine points.
"It is obviously January which is a tricky period for us historically," said Klopp. "You play without three players in Africa and a few others so we had a small group to get through this but the boys did incredible.
"It was a bit Jekyll and Hyde football today. It showed how insanely good we can be and how bad as well!"
Liverpool also have a game in hand to come on City and will have Salah and Mane back in time for their next Premier League game in 18 days' time.
City's 12-game winning run in the league was brought to an end in a 1-1 draw at Southampton on Saturday to offer Liverpool a glimmer of hope.
But Klopp believes Liverpool still have a long way to go to reel in the champions.
"The distance is too far to talk about it but we don't have to. We just need to win football matches," added the German.
"Each Premier League game is difficult like we saw today. We made it more exciting than necessary! We will see what happens in the final few months."
Palace boss Patrick Vieira was furious at his side's momentum being cut short by the penalty given for Vicente Guaita's block on Diogo Jota.
A lengthy VAR review was needed before the spot-kick was awarded at a time when Liverpool had been hanging on to their slender lead.
"When we scored, the game was there for us to go and try and get that draw. But the referee killed our legs today," said the former Arsenal captain.
"We had the momentum and the tempo. Then there was the penalty decision. The wrong decision by the referee had a massive impact on the game."