Sadio Mane made the most of some slack defending to score the only goal of the game after 73 minutes and secure the Reds' first Premier League win of 2017-18.
The home side were short of their best at Anfield and could have found themselves behind midway through the second half, with Christian Benteke spurning a glorious chance against his former club.
Klopp believes it was a difficult match given Palace's desire to sit back and praised his players for their performance.
"[It was] a hard job," he told Sky Sports. "I have no idea how much we had the ball but it was a lot. We had moments in the first half but had to be more patient because they were deep.
"We didn't score but we didn't give counter-attacks away. Second half, we did it better, the first part was really good. We were really awake and defended really well and I loved the goal because it was not a clear situation. It was a counter-pressing situation. We had other chances but I am completely happy with how did the job.
5 - Sadio Mane has had a hand in five goals in six Premier League games vs Crystal Palace (four goals, one assist). Thorn.— OptaJoe (@OptaJoe) August 19, 2017
"Anfield helps a lot. I think it was the difference."
Mane's goal came after a loose touch from Luka Milivojevic was blocked into his path inside the penalty area, leaving goalkeeper Wayne Hennessey – who had been excellent throughout – with no chance of making a save.
Palace boss Frank de Boer was frustrated to see their good defensive work undone by such an incident, but insists there were plenty of positive signs.
"We played almost 90 minutes very solid," De Boer, who endured a 3-0 home defeat to Huddersfield Town in his first league match in charge, told Sky Sports.
"We frustrated Liverpool sometimes and we were very organised. We played a little deeper but also that was because Liverpool are a team you have to respect.
"We have to fight every minute together. In 15 minutes we lost the game at Selhurst Park and we were 95 minutes very concentrated and [with] one unlucky touch we conceded the goal.
"I am happy with the spirit, not the result, but we can build from here."