Diogo Jota scored the winning goal as Liverpool came from behind to pass a stern test against Sheffield United.
Sander Berge's controversially-awarded penalty put the visitors, the better team for much of the first half, in front.
The Reds found an equaliser before the interval through Roberto Firmino and Jota's header on 64 minutes just about proved enough to secure the champions' first league win this month.
Story of the match
Liverpool went closest in the opening stages - John Egan taking the ball off the toe of Mohamed Salah in the goalmouth after Sadio Mane had knocked it past the impulsive Aaron Ramsdale - but Sheffield United nonetheless started strongly.
When Oli McBurnie went to ground under challenge from Fabinho, Mike Dean awarded the Blades a free-kick which would have had Liverpool sweating. But then came the signal in the ear from VAR, who took a second look at the tackle and decided it had taken place inside the area. Dangerous free-kick upgraded to penalty.
For the second weekend in succession, Liverpool had grounds for incredulity. A foul? Maybe. Inside the box? Surely not? McBurnie's forward tumble was perhaps deceiving.
Berge duly stepped up and sent the returning Alisson the wrong way, securing a precious lead.
If the decision left Jurgen Klopp dismayed, his side's initial reaction would surely have riled him even more.
Sheffield United took charge of the game. On 21 minutes, McBurnie - perhaps a fraction offside - took down Ethan Ampadu's ball and missed the target with a cross-goal effort from seven yards.
Moments later, Ben Osborn arrived to meet the delivery of opposite wing-back George Baldock on the volley and saw a sweet but central strike pushed away by Alisson.
They were worthy leaders even if the circumstances of the goal had been fortunate.
But Liverpool, after a steady spell of possession, began to show signs of improvement as half-time neared.
And the first injection of quality into their attacking play brought reward. Jordan Henderson whipped an excellent cross into Sadio Mane, and after his downward header was parried by Ramsdale, Firmino had the straightforward task of tapping-home the rebound.
Sheffield United, who creditably matched Liverpool in their desire, did not look fazed. They would fashion the first opportunity of the second half, with Berge's knock-down fiercely sent goalwards by Baldock before and whistling past the post via a deflection.
Liverpool thought they had tamed the visitors just after the hour mark, only for VAR to thwart them once again. Salah showed immense composure to juggle Trent Alexander Arnold's curled pass and volley it past Ramsdale, but on inspection, he had gone ever so slightly too early.
The most impressive aspect of this scruffy Liverpool performance, though, was that they refused to petulantly curse their luck. They quickly regained focus and when Mane's measured delivery was headed-in by Jota at the far post, there was no cause for intervention.
For the 10 minutes after that cherished goal, Liverpool were in control, poised to score another and cruise over the line.
As it transpired, with Salah denied by the post after spinning away from Egan, they could not stave off a nervy finish.
David McGoldrick lashed narrowly over and fellow substitute Oliver Burke bent a shot wide from 20 yards in stoppage time as the game flashed before observers' eyes.
Liverpool survived. A far from convincing display, but just how often do those end in victory for Jurgen Klopp?
Impact of Van Dijk absence becomes clearer
This was a game that further illustrated the extent of the challenge ahead. Liverpool will be without Virgil van Dijk for the vast majority, if not the entirety of the season. They were hardly bulletproof when he was playing this season, but now there is a new way to get at them.
The Dutchman's aerial dominance is unmatched, and that was often the undoing of sides who tried to hurt Liverpool in a direct fashion. But now attackers will win their headers much more frequently, as McBurnie did tonight, and following the knock-downs they can get at the Reds in numbers.
Joel Matip is more imposing in these situations than either Fabinho or Joe Gomez, but there are signs that his involvement will be very limited.
Sheffield United maintain principles and faith
Chris Wilder promised controlled aggression before the game and that was what Sheffield United fans got. Tellingly, he refused to play down the fixture's importance on the cliched grounds that it was not a fixture that would 'define their season'.
In terms of points, it has clearly been a concerning start. But United will not slump into the kind of relegation battle mentality which permits damage limitation exercises against top sides. Nor should they, based on a first half where they outperformed the reigning champions in each direction.
Positives are not points - indeed, you will struggle for points when you're so easily undone by crosses - but with Rhian Brewster, United's combination of spirit and (enhanced) quality should see them through a difficult spell. Do not write them off against Chelsea or Manchester City.
Man of the match - Jordan Henderson
An excellent cross to haul Liverpool level in the first half and a tireless contribution in the second, featuring plenty of gritty defensive work and a number of dribbling runs to ease the mounting pressure.