Mohamed Salah put the title winners on top in the sixth minute and just two minutes later, captain Jordan Henderson doubled their advantage.
Leandro Trossard halved the deficit in first half stoppage time before Salah scored his second of the game on 76 minutes to make it six wins from six against Brighton in the top flight.
Story of the game
Jurgen Klopp's men showed their class from the beginning and were ahead on six minutes when Naby Keita won possession from Davy Propper and squared for Salah, who finished past Mat Ryan.
Two minutes later, it was two after Adam Webster gave away possession to Keita, who was again the architect of the visitors attack as he found Roberto Firmino, who passed quickly to Salah. He spotted Henderson in support and the skipper gave Ryan no chance with a 20-yard thunderbolt.
The first attack for the hosts started when Neal Maupay charged down the left and found Leandro Trossard, whose effort was blocked by debutante Neco Williams. Maupay saw his own chance denied by Allison around the post.
Salah nearly completed his hat trick in the 38th minute as Keita's pinpoint pass found the Egyptian's feet, but he curled the shot wide of the Brighton post. On the stroke of halftime, the Seagulls found their way back into the game.
The impressive Tariq Lamptey crossed for Trossard, whose half-volley on the stroke of halftime made it 2-1. Ryan was equal to the task in stopping Salah, who had freed himself from Dan Burn and was played in once again by Keita.
Ryan was alert to the danger when Salah turned provider for Alex Oxlade-Chamberlain, who had his shot deflected away by the Seagulls goalkeeper with Henderson's rasping volley well-held by the keeper.
Burn should have equalized on the hour mark after Propper flicked Alexis Mac Allister's cross on to the left back, but Joe Gomez saw off the danger. Scrapping to stay close, the home side were soon undone by more Liverpool brilliance.
Salah found acres of space at the near post to cleverly head past Ryan and he could have completed his hat-trick in stoppage time, but headed over. Henderson was forced off after a collision with Yves Bissouma late on, but Klopp's men are the fastest top-flight team to 30 wins in a season as well as nine points from setting a new points record.
Man of the match: Mohamed Salah
Both of his goals once again showcased the striker's class, his power on display for the first and his touch on the second with his head. He now has 19 goals for the season and is etching his name amongst the legends at the club.
Liverpool's attack was at its best since winning the title
After being thrashed at Manchester City and turning in a lackluster effort against Aston Villa, the Reds were sharp in attack with Keita the primary creator. Salah was his usual brilliant self and Henderson's goal added a touch of class as the champions will be encouraged by this display in pursuit of the points record.
Brighton showed why they're staying up
After going two goals down so quickly, many teams would have folded against Liverpool, but Graham Potter's men showed why a fourth successive season in the top flight is on the cards. The fightback, determination and graft produced against the champions is a staple of the former Swansea City boss' side as he has established a work ethic and confidence at the club.