Manchester City's 2-2 draw at Newcastle United earlier in the day gave Jurgen Klopp's men the chance to extend their healthy lead over last season's champion to 11 points, a task they looked set to complete with ease after Virgil van Dijk scored twice inside 24 minutes at Anfield.
Brighton was given an unexpected lifeline when goalkeeper Alisson was sent off for handling outside his box, meaning he will miss Thursday's (AEDT) Merseyside derby, and Lewis Dunk caught out replacement goalkeeper Adrian with a quick free-kick.
But the Seagulls could not find a late equaliser, and while Leicester City can cut Liverpool's advantage at the summit to eight points by beating Everton on Monday (AEDT), there looks to be no stopping a team that has now gone 31 league games unbeaten, matching a record streak from May 1987 to March 1988.
A Trent Alexander-Arnold free-kick set up the opener for Van Dijk in the 18th minute and the same combination soon provided a second goal as Brighton was found wanting at a corner.
After a quiet start, Liverpool was looking rampant and Australian Mat Ryan made a trio of fine stops as Sadio Mane laid on two chances for Roberto Firmino before being denied himself.
Dunk should have pulled a goal back eight minutes before the interval during Brighton's best spell, but the defender clumsily shinned the ball wide from Dan Burn's cross, having perhaps been put off by Yves Bissouma failing to get fully out of the way from a clearly offside position.
After heading wide from a corner early in the second half, Dunk made it third-time lucky by drilling home a free-kick with Adrian out of position, after Alisson had used his hands outside the area to deny substitute Leandro Trossard.
Liverpool was on edge thereafter and Adrian saved well from Ryan's Socceroos team-mate Aaron Mooy before almost spilling a tame Pascal Gross header into his own net as the hosts just about clung on.