The Blades started like a team wanting to impress new manager Chris Wilder, and James McAtee really should have put them in front inside 12 minutes, but saw his tame shot saved by Caoimhín Kelleher from close range.
That seemed to wake Liverpool up, and the visitors started to threaten, particularly from set-pieces. Virgil van Dijk saw his header tipped over by Wes Foderingham before finding the net from another Trent Alexander-Arnold corner, as the Dutchman volleyed in his first goal since March.
It means Sheffield United are still searching for their first clean sheet this season, and that goal was the 40th they have conceded in just 15 matches - the most-ever at this stage in the PL era.
Liverpool’s set-piece threat continued into the second half, as Mohamed Salah was denied his 200th goal for the club by a brilliant Foderingham save, after the Egyptian volleyed another perfect Alexander-Arnold delivery.
The hosts, who were the lowest scorers in the division before tonight, did not really threaten a response, and Jürgen Klopp felt that he could take top-scorer Salah off midway through the second period.
His replacement was Darwin Núñez, who was desperate to get on the scoresheet having gone six games without a goal before tonight, but the striker saw his late effort saved in an attempt to make the scoreline more comfortable.
Deep into injury time, Dominik Szoboszlai did just that, as he rifled home in acres of space in the penalty area to send the away support wild.
A crucial three points for the Reds in terms of the title race, who have now remarkably won their last 10 PL matches played on a Wednesday.
Sheffield United have now lost their last six H2Hs against Liverpool, scoring only one goal - leaving them rock-bottom of the table with Wilder anticipating a lot of work to do to save their top-flight status.
Flashscore Man of the Match: Virgil van Dijk (Liverpool)
