Ten-man Chelsea withstood second-half pressure to earn a point at Anfield.
Kai Havertz put them ahead midway through the first half with a superb back-header from Reece James’ left-wing corner.
But the Blues were dealt a double blow just before the interval when James’ handball led to him being sent off and Mohamed Salah equalising with the resulting penalty.
The visitors were on top up to that point, with Havertz diverting James’ delivery into the far corner of the net.
However, during a goalmouth scramble in which Joel Matip headed against the bar, Sadio Mane’s follow-up was blocked on the line by James, with the ball ricocheting off his thigh onto his arm.
After checking the pitchside monitor, referee Anthony Taylor awarded a penalty and showed James a red card.
Inevitably, Liverpool piled forward in the second half in an attempt to capitalise on their numerical advantage.
But Chelsea defended brilliantly to preserve their unbeaten start to the season.
And Mateo Kovacic missed a late chance to put them ahead when he shot straight at keeper Alisson after combining with Romelu Lukaku.
Chelsea: Mendy; Azpilicueta, Christensen, Rudiger; James, Jorginho (Chalobah 87), Kante (Kovacic 45), Alonso, Mount, Havertz (Silva 45), Lukaku.
Subs not used: Kepa, Werner, Loftus-Cheek, Hudson-Odoi, Chilwell, Ziyech.