Mohamed Salah's late equaliser helped cancel out two goals from Aleksandar Mitrovic, with Darwin Nunez having earlier levelled for the Reds after Fulham's opener.
Serbia striker Mitrovic looked every part the player who smashed Championship records with 43 goals last season when he opened his account for the campaign in the 32nd minute, before following it up with a second-half penalty.
Nunez had followed Mitrovic's first with a clever improvised backheel goal, before turning provider on his league debut for Salah, who slotted home Liverpool's crucial second 10 minutes from time.
Despite rescuing a point, Klopp believes his side were far from their best at Craven Cottage.
"We started exactly the opposite way than we wanted," the Liverpool boss said.
"We are used to more dominance and we couldn't get that because we played just not good enough, not quick enough, not precise enough, all these kind of things.
"I'm really fine with the point. I know we could have won the game - that wouldn't have made our performance a little bit better, but we could have.
"I'm happy for the punishment we got with losing two points. I know we can see that differently - for them (Fulham) I'd say as well they lost two points."
The warning signs were there for Klopp as Liverpool struggled to get going and string passes together early on, before being caught on the break.
Although they grew into the game, it was far from a composed Reds performance.
"I feel much more responsible for defeats than when we win, so that's unfortunate," Klopp added.
"The performance was a defeat. The result is not a defeat but the performance was a defeat.
"So that's why we have to figure out what happened. That's not us today.
"But we don't want to take anything away from Fulham, they did really well."
Marco Silva believes his Fulham side's performance merited more than the point they came away with against a side who reached the Champions League final in May.
"I think that our players deserve something more than just one point," the Fulham manager said.
"They had more control of the game, but I think that our organisation, our great first half, the way we blocked almost everything from them, in terms of organisation in the first half - it was almost perfect.
"In terms of intensity the first half was really good and I have to congratulate our players and our fans as well."
When asked about the performance of Mitrovic, he added: "In the box he will score if you provide him good chances.
"But I repeat something I said last season maybe 20 times: Mitro is not just a goalscorer for us. He did an unbelievable job for us without the ball."