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Diogo Jota's second-half header earned Liverpool a first win in three league games to lift them joint top with Merseyside rivals Everton, who play on Sunday (local time).
Manchester City's stuttering start continued as they drew 1-1 at West Ham United, while Manchester United and Chelsea played out a drab 0-0 draw at Old Trafford.
Crystal Palace moved up into the top five as they beat bottom club Fulham 2-1 with goals by Jairo Riedewald and Wilfried Zaha.
Liverpool had not conceded a penalty at their Anfield fortress for 67 Premier League games. However, that run came to an end early on against Sheffield United as Fabinho was adjudged by VAR to have fouled Oli McBurnie inside the area, even though referee Mike Dean had awarded a free kick outside the box.
Sander Berge sent Alisson, back in goal after injury, the wrong way to put the visitors ahead and Liverpool lived dangerously before Roberto Firmino levelled in the 41st minute.
Jota's header earned the three points but it was a struggle in the rain for Juergen Klopp's side who looked uneasy at the back without the injured Virgil van Dijk.
For the second week running they also had legitimate gripes about VAR.
"It's nice to win 2-0 or 3-0 every game but that only happens if, in games like this, we are ready to fight hard," Klopp said. "On nights like this you earn the right to win the others like that.
"The penalty was not even a foul. During a season there are so many different periods, we have had an injustice and we have had to go again."
Manchester City trailed to Michail Antonio's acrobatic opener at the London Stadium and only improved after the break when youngster Phil Foden came on for the injured Sergio Aguero.
Foden grabbed the equaliser and Raheem Sterling wasted a glorious late chance as City were frustrated.
They are in 12th place with eight points from five games, their worst start to a Premier League season for six years.
"I'm not a guy who can predict the future. We've already dropped seven points, which is a lot. But we've struggled a lot for many reasons," said City manager Pep Guardiola, adding that the recently-returned Aguero could have a hamstring injury.
While City languish in mid-table, neighbours Manchester United are down in 15th with seven points from five games after their stalemate against Chelsea.
United manager Ole Gunnar Solskjaer said both teams were showing the effects of their midweek Champions League exertions -- not helped by an empty stadium. Although United were able to call on Edinson Cavani for his debut.
"It was two teams who played in Europe this week and it showed," he said. "If that Stretford End was full we might get that ball over the line. It is a different sport with the fans here. We miss them. It's the same for all teams but everyone agrees there's a little less in it -- the edge has gone."
United avoided a third consecutive home league defeat at the start of a season for the first time since 1930, although they were fortunate when Harry Maguire got away with what looked like a foul inside the area on Cesar Azpilicueta in the first half.