Liverpool hung on after having goalkeeper Alisson sent off to beat Brighton & Hove Albion 2-1 and extend their lead at the top of the Premier League to 11 points thanks to two first-half headers from Virgil van Dijk.
Liverpool, who also equalled their unbeaten top flight run of 31 games set between May 1987 and March 1988, have 40 points from 14 games, with second-placed champions Manchester City on 29 after they were held to a 2-2 draw at Newcastle United.
It was a nervy record-extending 14th successive Premier League win at home for Liverpool, who missed a string of chances either side of Van Dijk's double before they were forced to defend desperately in the closing stages.
The towering defender acknowledged Alisson's sending off was a blow ahead of Liverpool's Merseyside derby at home to Everton on Wednesday, which he will miss through suspension, and also praised Brighton.
"We should have done better," Van Dijk said.
"We put him in that situation but it is what it is. We have to deal with it. It is a disappointment but we go again.
"They [Brighton] are a good side. They play football and take risks. It's not easy. We should have gone better than we did in the second half but you have to give them credit as well."
Van Dijk opened the scoring in the 18th minute with a looping header after Trent Alexander-Arnold floated in a free kick from the right and the two combined again for the second goal six minutes later.
Brighton goalkeeper Mat Ryan denied Sadio Mane with a fine reflex save but was powerless when Van Dijk made his run to meet Alexander-Arnold's corner at the near post.
Centre back Lewis Dunk missed two chances to pull one back for Brighton, sending a shot wide from close range shortly before halftime and then glancing a header inches wide of the far post in the 54th minute.
Then came a frantic climax as Alisson was sent off in the 77th minute for handling the ball outside the penalty area.
While stand-in keeper Adrian was still yelling out instructions to the wall at the far post, Dunk took advantage and stroked a quick low free kick into the opposite corner.
Brighton pressed for an equaliser and Adrian parried an Aaron Mooy piledriver in the 85th minute as the home side held on, much to the delight of the packed crowd at Anfield.