Curtis Jones' close-range finish just before the hour mark proved the difference, though the scoreline belies the open nature of the game.
The Reds prevailed even in the face of nine first-team injuries, with Alisson the most recent addition to the list after picking up a hamstring issue.
Jones deservedly an Anfield regular
Phil Foden, Mason Greenwood, Bukayo Saka. You can add Curtis Jones to that list. The latest English attacking talent is progressing from star of the future to star of the present.
The 19-year-old became the third youngest European scorer in Liverpool history with his match-winning strike, intelligently anticipating the misjudgment of Andre Onana to turn home.
Earlier, he had been denied a far more spectacular goal by the post, having connected sumptuously with a lay-off to the edge of the area.
Jones has exuded confidence ever since his debut, but his maturation in recent times is striking. He is a pleasure to watch, particularly when he receives the ball on the turn between the lines.
This was eighth start of the season, and while this is partly down to the mere fact that he has stayed fit, he will surely remain a regular even when the Reds' injury crisis abates.
Unflappable Kelleher primed to leapfrog Adrian
This was a daunting test for Caoimhin Kelleher. Having shipped seven (seven) to Manchester City's Under-23s last weekend, he was tasked with thwarting an attack which had a staggering 42 league goals under its belt after just 10 games.
But remarkably, there were no signs whatsoever of nerves. Kelleher commanded his penalty area well and distributed the ball assuredly.
In truth, he would have been expected to save all of the shots on target he faced, including the menacing Noussair Mazraoui long-ranger and last-gasp Klaas-Jan Huntelaar header.
But how often has Adrian been beaten when he ought to have done better?
Kelleher has, indeed, tied down the goalkeeping spot for the duration of Alisson's latest spell on the sidelines.
The 22-year-old Irishman suddenly has the opportunity to become understudy to one of the best 'keepers in the world.
Williams' gutsy display
And that was before a clumsy challenge at the weekend handed Brighton a penalty and contributed to his half-time withdrawal.
He may have been privately concerned, but he got a solid performance - diligent defending and an important attacking contribution when the Reds at last started to mount assaults down the right.
He whipped a cross into uncertain territory and it evaded Onana, presenting the golden chance to Jones.
More impressive than the technicalities was the show of character. Williams looked like a man who had retained total focus under the microscope, and that is to be commended.
This is something to build on.
Injury-ravaged Reds have secure foundation
Since they lost the best defender in the world to injury, Liverpool have conceded only six goals across 10 matches.
Remarkably, that represents an improvement on what came before.
Fabinho is playing deeper but he has retained his trademark aggression. With every superbly-timed, clean tackle on the edge of the area he looks an increasingly accomplished centre-half.
Matip, meanwhile, remains an exceptionally solid defender in spite of his persistent fitness struggles.
Whisper it (because of the unrelenting succession of injuries) but Liverpool look to be in safe hands.
A well-earned luxury
At last, Jurgen Klopp and Liverpool have breathing space.
They are not back in action until Sunday evening when Wolves visit Anfield, opening up invaluable room for recovery and preparation.
And, with the privilege of a runner-up opponent in R16 already secured, they can afford to leave a host of key players at home for their journey to Denmark in eight days' time.
The first XI will effectively enjoy a week's rest for the first time since late September.
They will therefore be relatively well-placed for the looming festive fixture onslaught.
The importance of this victory extends beyond qualification.