The runaway Premier League leaders needed an 85th-minute winner from Roberto Firmino to maintain their eight-point advantage at the top of the table, with the Brazilian quickly cancelling out Wilfried Zaha's equaliser.
The result means that Liverpool have now picked up 37 points from the 39 on offer so far this season, and Klopp acknowledged that it has been an "exceptional" start from last term's runners-up.
"I'm not surprised at all it was tough. We know it's a tough place, the atmosphere is good, they don't give up. These players don't need a lot of situations to stay in the game, we gave them much too many today because we were not compact enough. We had the challenges in the wrong areas, which meant when we blocked them it was always a set piece, which helped them more than us," Klopp told reporters at his post-match press conference.
"The start was good and after that we had really good football moments. Crystal Palace as well, we knew they could play good football, but more often direct, which they did pretty well. They had moments on set pieces and we knew at half time that we had to do better. I said to everybody and their dogs that the challenge is they come back from so many different national teams. Today I have no problem that we were not brilliant because in a game like this you just have to make sure that you are ready to fight for the result and we did that from the first minute.
"Could we have played better? Yes. But we had players who were still on a plane back from [international duty] on Thursday morning. That's not too cool. Getting a result of Crystal Palace we'll never take for granted so it feels good. 37 points after 13 match days is exceptional, and you cannot have them winning only brilliant games. We had a couple of brilliant games and we won them, thank God, but we can't only win them. We are not out there to send messages, we just want to get results.
"I really don't take it for granted - how can you? With all the things today where we were not so good we know that we were the dominant side, we were the much more active team, Crystal Palace defending deep most of the time and we don't stop with those things even when it's not perfect.
"So when we concede a goal, you see the faces of the boys - we know a set piece, a corner, a free kick is a chance. I thought the ball was in twice before Bobby finished it off, but we scored that goal and from that moment on we defended really well. There was a bicycle kick from Christian [Benteke], but apart from that not many chances with five minutes to go, maybe 10 with injury time. We did well in that time."
Liverpool are now unbeaten in their last 30 Premier League games, the joint-third longest ever run in the competition's history.