The Reds’ display wasn’t completely convincing, but goals from Virgil van Dijk and Dominik Szoboszlai secured three points to maintain their Premier League title bid.
Jurgen Klopp’s men dominated possession but failed to create many clear-cut chances in the first half.
Ibrahima Konate had a good opportunity to break the deadlock, but his weak header was easily saved by Wes Foderingham.
The Blades were dangerous on the break, and James McAtee came close to scoring after he ended up one-on-one with Caoimhin Kelleher.
Van Dijk gave Liverpool the lead after 37 minutes, with the defender meeting Trent Alexander-Arnold’s cross with a brilliant volley to find the bottom-left corner.
Mohamed Salah, just one goal away from scoring his 200th goal for the Merseyside club, could have doubled their lead but he scuffed his shot in the box.
Alexis Mac Allister produced another brilliant effort just inside the box in stoppage time, but Foderingham tipped the ball over.
Liverpool went into the break with a 1-0 lead, but their performance didn’t suggest they were in complete control of the match.
Salah made a good effort to reach the 200 mark after 56 minutes but the Blades’ keeper denied him with a fantastic stop.
The Egyptian was taken off the field around ten minutes later by Jurgen Klopp, who also withdrew Mac Allister after the midfielder picked up a knock.
United had a massive chance to get back in the game after a great ball from McAtee but Cameron Archer wasted the opportunity.
Nunez then went on the attack but his effort came to nothing after a smart save from Foderingham.
The home side pushed for an equaliser, but Szoboszlai sealed victory for Liverpool by side-footing the ball home from close range in the fourth minute of added time.
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