Liverpool stormed to 5-2 victory over lowly Everton at Anfield, with Klopp bringing up his 100th Premier League win in the process.
It also marked a new club record, as Liverpool extended its unbeaten run in the top flight to 32 matches. It was last defeated by Manchester City in January.
Divock Origi scored twice to take his tally of derby goals to five, while Xherdan Shaqiri capped his first start of the season with a goal - regulars Sadio Mane and Georginio Wijnaldum completing the rout.
"I speak a lot about the quality of the squad so I have to show it from time to time, but then the boys have to show they can perform," Klopp, who made five changes, said. "It was incredible – Divock's goals, Sadio's passes, Shaq's linking play, everything, Adam Lallana was everywhere and we had so many really good moments.
"But of course a bit rusty in defending, they were really direct. [Everton] couldn't defend us but offensively they caused us problems at times. All good at 4-1, the goal for 4-2 I didn't like too much because the injury time is three minutes and we start playing an offensive pass. But we have to learn, have to improve and we will do.
"The squad didn't become bigger tonight, we knew these boys were here, we knew they were fantastic footballers.
"This was the first game of the tough fixture list, we have to make changes and will make changes. I'm really pleased, really happy but I was not in doubt about it. Of course not for me really, but for the public, for the crowd, they feel much better about the squad."
Silva's chances of keeping hold of his job at Everton, meanwhile, seem increasingly slim, with the Toffees sitting 18th after 15 games.
Everton is in the relegation zone after 15 rounds for the first time since 1999, though the former Watford boss insisted he is still focused on attempting to turn things around.
"I am not the right person to talk to you about the situation. I cannot answer you," Silva said. "Since I joined the club and until the last day I am here I will always be 100 per cent professional, giving my best every single day.
"I understand why the questions are coming. If we are winning all the games, you are not asking them. It is not for me to take those situations. I don't like to talk after every single match about my future."