Curtis Jones and a Kean Bryan own goal ended the Reds’ run of four successive League defeats and heaped more misery on the bottom-placed Blades.
Bryan’s own goal, which was originally credited to Roberto Firmino, was Liverpool’s 7,000th goal in top-flight football and secured an important win for Liverpool.
With West Ham and Leicester City losing this weekend and Chelsea drawing with Manchester United earlier on Sunday, Liverpool sealed three valuable points in the race for Champions League qualification to move within two points of fourth-placed West Ham.
Sheffield United’s 21st defeat from 26 games leaves them 15 points from safety with a seemingly impossible battle to beat the drop.
And in a tribute to team-mate Alisson Becker, who missed the match following the death of his father Jose in Brazil, Liverpool players wore black armbands while Adrian made his first Premier League appearance since the Merseyside derby at Everton on October 17.
Additional reporting by PA.