Liverpool took the lead after four minutes when Andrew Robertson capitalised on a slip from George Baldock to cross for Salah, who shot underneath Dean Henderson.
The Egyptian was denied twice by Henderson before half-time with Liverpool firmly in control.
Salah hit the post early in the second half before Mane made it 2-0 in the 64th minute.
Mane played a one-two with Salah before being denied by Henderson, but he scored at the second attempt into the roof of an empty net.
Alisson saved from Sheffield United substitute Oliver McBurnie's close-range effort late on as Liverpool secured an 11th successive league victory to stay unbeaten for more than a year.
Sheffield United remain eighth with 29 points.
Liverpool: 11 Jan v Spurs (A)
Sheff Utd: 10 Jan v West Ham (H)