Liverpool had to do it this time without injured defensive talisman Virgil van Dijk and at the end of a demanding week in which they drew 2-2 at Everton and ground out a 1-0 win over Ajax Amsterdam in the Champions League.
"A tough week obviously -- a lot of games, a lot of minutes, tough opponents, hard-fighting opponents," manager Jurgen Klopp said after the match.
"[We] won the two matches after the Merseyside derby, so really pleased with that, obviously.
"We didn't score [the third goal] so the game stayed open and they had their moments, but we defended well with a big heart. And so I liked it."
The visitors took a surprise early lead from the spot after VAR awarded Sheffield United a penalty against Fabinho and Sander Berge sent Alisson the wrong way to make it 1-0 inside of 15 minutes.
Alisson did well to keep the deficit at a single goal with an excellent reflex save, parrying away Ben Osborn's smashing first-time volley in the 24th minute.
Mohamed Salah looked to have put Liverpool in front just past the hour mark, but VAR once again intervened and the goal was ruled out for offside on the Egyptian as the match stayed level.
Minutes later, the hosts did retake the lead when Mane played a perfect cross from the left flank and Diogo Jota elevated to head home at the back post.
It wasn't vintage Liverpool football, but captain Jordan Henderson said that was to be expected playing against a well-coached team like Chris Wilder's Sheffield United.
"I am only interested in the mentality in our changing room and I see hunger, resilience, character to dig in and win if you are not playing at your best," Liverpool skipper Jordan Henderson said. "We showed we can dig in and see results out."
Salah should have extended Liverpool's lead in the 80th minute after his clever turn at the top of the area left him with just Ramsdale to beat, but his close-range shot came back off the near post and the scoreline stayed 2-1.
Jota has now scored in his first two home league games since joining from Wolverhampton Wanderers.
Incredibly, since netting against Arsenal on Sept. 28, the Portuguese has played in eight games for club and country.
Klopp said there was more to come from the 23-year-old.
"He is still adapting. In the moment, a lot of things we do are not natural to him but that's only because Wolves play different -- but a lot of things he did for Wolves are exactly what we want him to do," Klopp said.
"That he is that close already is just a sign of how good a player he is. As I said, if he stays fit, he's 23 and the future is bright. We will need his quality and I am really happy that he's settled nicely so far."
The result puts Liverpool level on points with Everton atop the Premier League, although Carlo Ancelotti's side lead on goal difference and can retake sole possession of first with a win against Southampton on Sunday.
Sheffield United are stuck in the relegation zone after having managed just a single point from their six league games this season.