Salah put Liverpool ahead after just four minutes thanks to good work from Virgil van Dijk and Andy Robertson. Van Dijk played an exquisite ball over the top for Robertson, who slid it into the box for the Egyptian.
The Reds forward poked the ball through the legs of Blades keeper Dean Henderson to get the league leaders off to a perfect start at Anfield.
The hosts were denied by Henderson soon after as he somehow managed to deny Salah a second. Jordan Henderson and Georginio Wijnaldum were involved in the build-up and the former picked out Salah with a whipped cross into the block. The 27-year-old hit a sweet volley, but Henderson somehow managed to get a hand to the shot and claw it over the crossbar.
United’s John Lundstram saw a goal disallowed for a clear offside on 15 minutes, while at the other end Salah was denied again by Henderson on the half hour mark.
Liverpool cruised through the first half but stepped up a gear after the restart and Van Dijk should have scored from Robertson’s cross but failed to make contact.
Roberto Firmino the fired wide before Jurgen Klopp’s men got their second on 64 minutes through Mane with an impressive team move that started with Alisson.
The keeper released it quickly to Robertson and he then sent Sadio Mane on his way down the left.
Mane and Salah played a one-two to take out the Blades’ centre-backs.
The Senegal frontman’s initial effort was blocked by Henderson but Mane was able to slam home from a yard out after the ball rebounded back to him.
Mane and Firmino both had chances to grab a third, but Liverpool had to settle for a 2-0 win.
Liverpool restored their 13 point lead over closest rivals Leicester at the top of the Premier League table with the win, with Manchester City a point further back in third.
The Blades remain in eighth spot with 29 points after 21 games.