David Luiz's quick thinking with a sublimely executed free-kick put Antonio Conte's men ahead in the 24th minute, but Liverpool were in the ascendency after Georginio Wijnaldum headed an equaliser 12 minutes into the second half.
It looked as if their efforts would be in vain with a quarter of an hour remaining, but Simon Mignolet kept out Diego Costa's penalty in front of the Kop.
If Chelsea's top scorer had converted it would have been four consecutive home losses for Liverpool, who have bowed out of the EFL and FA Cups during that run.
Nevertheless, Klopp's team remain firmly within the chasing pack in the top flight, only a point behind second-placed Tottenham, and the German was buoyed by Tuesday's performance.
"I'm not sure that happy is the right word, but I am proud of the performance because the boys did well against a really strong side," he told BT Sport.
"In a few moments, sure, they were a threat, but they were not as strong as they usually are – that is because of my team, we did really well.
"We gave a very important signal for our fans that we are still there.
"Yes we lost games and yes we didn't play good football. I think this wonderful powerful club still needs to improve and keep our nerve. Not everything is bad because you lose a football game.
"This team is outstanding in attitude. Let's do the best we can. We will fight for each point. Everybody saw it."