Georginio Wijnaldum's 70th-minute goal settled the affair to ensure the Reds' lead at the top of the Premier League will be at least five points at the end of the weekend.
Here are five talking points from the match...
Watch: Sheff Utd 0-1 Liverpool in 90 seconds
Sixteen and counting
Liverpool extended their Premier League winning streak to 16, but had to work extremely hard for the three points in Sheffield.
It means the run now stands just two short of the record set by Manchester City in 2017.
The Reds were given a fair number of scares by Chris Wilder's side and Wijnaldum's goal was the visitors' first shot on target in the match.
After Divock Origi's cross was cleared to the edge of the penalty area, Wijnaldum fancied his luck with a volley towards goal that Dean Henderson couldn't keep out.
It won't be the cleanest goal Liverpool score in 2019-20, but the resulting celebrations told you of its importance.
If you don't buy a ticket, you can't win the raffle...
Like they did at Milton Keynes Dons in midweek, Liverpool again benefited from a goalkeeping error to break the deadlock.
Wijnaldum's strike went through the legs of the impressive Henderson and trickled over the goalline, and that was the difference at Bramall Lane.
As the manager regularly points out, luck plays a huge part in the determination of football matches. Liverpool proved that here.
Newcomers beware
Welcome to the top flight.
This result means Liverpool have now won their last 13 Premier League matches against newly-promoted sides. In that time they have scored 36 goals, conceding a mere four.
They handed last season's Championship winners Norwich City a 4-1 defeat on the opening day and inflicted misery on the runners-up here.
The Reds face the last of the promoted teams on November 2 when they travel to Aston Villa.
Solid at the back
Virgil van Dijk and Joel Matip impressively stonewalled United's attackers, heading and clearing anything that came towards Adrian's goal.
But their overall contribution wasn't solely defensive. Indeed, against such a compact and determined opponent, the centre-halves' ability with the ball often unlocked the opposition.
In the first half, Matip found Wijnaldum in between the lines before unleashing a shot that was deflected for a corner. Then Van Dijk sent a perfect lofted ball in behind United's backline to find the rushing Mane, who ended up firing wide.
It's a partnership that continues to flourish at both ends of the pitch.
The Reds keep on going
Twenty-one points from a possible 21. Liverpool have become only the second team to win their opening seven Premier League matches.
They follow the trail of the 2005-06 Chelsea team that ended up winning the title after starting the campaign with nine straight victories.
Leicester City are the next opponents in the league for Klopp's side. But before that, the Reds must navigate Champions League action in midweek as they welcome Salzburg to Anfield and look to respond to the Napoli defeat.