Brighton and Hove Albion fought back bravely but it was poor defending from two set-pieces that cost them dearly at Anfield.
Two assists from Trent Alexander-Arnold and two headers from Virgil van Dijk gave Liverpool a comfortable 2-0 lead in the first half. Lewis Dunk made it interesting at the end with a clever free-kick after home keeper Alisson was sent-off for handling outside the box.
It was an improved away performance from Brighton who were doing okay until Alexander-Arnold found his range.
Maty Ryan had been forced into a couple of early saves but you expect at Anfield. On the whole, Albion kept possession well and Aaron Connolly - back in the team following a groin injury - was a real pest to home defenders Dejan Lovren and Virgil van Dijk.
But on 17 minutes Pascal Gross needlessly clattered into Alex Oxlade-Chamberlain on the right hand side. Dale Stephens was livid with Gross while Alexander-Arnold stood over the ball.
The young England defender then pinged the perfect delivery into the box and van Dijk rose easily, without a challenge, and nodded over Ryan.
Six minutes later and Liverpool were cruising at 2-0. Davy Propper conceded the corner and once again Alexander-Arnold picked out van Dijk's near-post run and the Dutch defender headed home with minimal fuss.
It all looked too easy, mainly because it was too easy. Albion defenders looked puzzled and Graham Potter, who before the match agreed an improved two-year contract extension that takes him to 2025, looked at the floor of his technical area and gently shook his head.
Brighton have struggled against the top teams on the road this season. Chelsea was a 2-0 battering, they played quite well at Manchester City but lost by four and at Manchester United they were ripped apart by the pace of Marcus Rashford and Dan James.
A combination of Liverpool easing off and Brighton with nothing to lose saw Albion enjoy a spell of pressure and Klopp's men began to look like a team that hadn't kept a clean sheet in their previous 11 league and cup games.
Propper had a left-footed effort saved by Allison. The corner was half cleared and Dan Burn's cross fell for Dunk but his first time effort from 10 yards bobbled off his shin and trickled wide.
Brighton finished the first 45 well but Liverpool have not lost in their last 97 Premier League games at Anfield when leading at half-time.
Albion stuck with their five man midfield after the break and they needed something special against a Liverpool team determined to capitalise on Manchester City's dropped points earlier today.
Connolly had a spin and a shot well saved, while Liverpool almost made it three when Alex Oxlade-Chamberlain dragged an effort wide after a swift combination with Alexander-Arnold.
Leandro Trossard, Steven Alzate and Neal Maupay all came on for Brighton as Potter played his full hand.
Trossard was soon in the action as he latched on to a long through ball from Adam Webster. The Belgian nipped ahead of Allison and Trossard's lob was handled by the keeper outside the box. Allison was sent-off and replaced by Adrian.
Adrian was still lining up his wall when the referee blew his whistle and the quick-thinking Dunk cleverly place his freekick into the bottom corner. Liverpool complained but the referee clearly blew his whistle to re-start.
Brighton sensed a point and Liverpool were wobbling. Aaron Mooy drove forward and unleashed a powerful effort that was beaten away. Adrian then made a hash of Gross' header but frantically gathered at the second attempt as Maupay lurked.
The whistle went, Dunk fell to the Anfield turf in despair and Klopp puffed out his cheeks in relief. It was a brave fightback from Brighton who produced an improved away performance but it was a lack of marking from Alexander-Arnold's two precise assists and van Dijk's two headers that killed them
Brighton: Ryan, Montoya (Alzate 67), Webster, Dunk, Burn; Stephens, Mooy; Gross, Propper, Bissouma (Trossard 67); Connolly (Maupay 75).
Subs not used: Button, Duffy, Murray, Schelotto.