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.
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.
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.