Mohamed Salah's late equaliser ensured it was not worse for the Reds, helping cancel out two goals from Aleksander Mitrovic, who had twice given Fulham the lead.
Substitute Darwin Nunez followed Mitrovic's first with an improvised backheel before turning provider for Salah, who tapped home for Liverpool's equaliser 10 minutes from time.
The Uruguayan earned the praise of his new manager, after scoring in the Charity Shield against Manchester City as well as the first league game of the season.
"A great goal, (he) could have scored another one as well, maybe two. Crazy," the Liverpool boss said.
"So he is involved in a lot of goalscoring situations and that's really cool."
Liverpool were far from their best at Craven Cottage, and Klopp admitted he will have to look into what went wrong on the banks of the Thames.
"I would have loved to get three points but I would have loved much more to play really good, to be honest, and we didn't do that. That's why I'm now in the mood," the German said, comparing the performance with the previous week's Community Shield defeat of Manchester City.
"It's my responsibility to find out why we played like that last Saturday (and why) we look like completely upside down (today). That doesn't make sense and that's my responsibility.
"So we may need to improve, figure out what went wrong."
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 with two goals against last season's runners-up.
Although the forward has previously had limited impact in the top flight - he scored just three goals in 2020-21 - Marco Silva does not believe that will be the case this season.
The Cottagers boss said: "Mitro is the top scorer of his national team and national teams do not play just against Championship clubs. They play against the best national teams in the world.
"He will be at the World Cup, he was decisive in the qualification play-off against Portugal. So, for me, some of the questions don't make sense.
"Mitro has his own profile. It's up to us to deliver and get the best from him. Of course, in some games he will score like this afternoon and in others he will not. Then he has to do the other part of the game as well."