But Alisson’s red card – for clearly handling outside his penalty area with Brighton substitute Leandro Trossard bearing down on goal – gifted the visitors a lifeline.
Substitute keeper Adrian’s first touch was to gather the ball out of his own net after Lewis Dunk’s cute free-kick left the keeper – who was still organising his wall – stranded, before Liverpool’s stopper saved late on from Aaron Mooy.
Earlier in the day, reigning champions Manchester City were held to a 2-2 draw at Newcastle United.
The win means Liverpool have equalled their longest ever unbeaten run in the top flight, having also gone 31 games without defeat between May 1987 and March 1988.
Yet this was anything but straightforward for the leaders as Alisson became the first Red keeper to be sent off in a Premier League game since fellow Brazilian Doni in 2012.
With his side 2-0 up and closing in on three points, Alisson came rushing out of his penalty area to handle the ball as Brighton attacked.
As a result, Alisson will now miss the Merseyside derby with Everton at Anfield on Wednesday.
His actions set up a nerve-wracking finale and Dunk’s free-kick – an awful goal to concede from Liverpool’s point of view – means Klopp’s side are still without a clean sheet since 28 September.
Substitute keeper Adrian saved his side by keeping out Mooy as mid-table Brighton piled forward in search of an equaliser.
The Reds had looked like coasting to victory after Van Dijk’s double, and Brighton keeper Mat Ryan made three fine saves to frustrate Roberto Firmino, twice, and Mohamed Salah.
Liverpool head into a demanding December, a month which will see the club play nine fixtures in four different competitions, with a fifth successive league win – but this was far too close for comfort.
