Virgil van Dijk's goal was his first of the season for Liverpool
Wilder, who led Sheffield United to two promotions in a five-year reign before an unceremonious departure in 2021, received a raucous reaction from the Blades faithful on his return to the dugout.
However, Virgil van Dijk swept home a side-footed volley from Trent Alexander-Arnold's corner to quell the vociferous home support despite Blades appeals he had fouled Anel Ahmedhodzic prior to doing so.
There was also an element of controversy about the Reds' second goal in stoppage time as Darwin Nunez's hefty challenge on Jayden Bogle was deemed fair in the build-up to Dominik Szoboszlai's cool finish.
Liverpool, beaten just once in the league this season when their nine men lost in stoppage time at Tottenham, move back to within two points of the table-topping Gunners.
The Blades, meanwhile, remain bottom, four points from safety, after a 12th loss in 15 league games, although Wilder will take heart from a competitive performance in defeat against the second-placed Reds.
"It was a very mature and resilient performance," Reds boss Jurgen Klopp told BBC Match of the Day. "We didn't let the atmosphere really happen. I think everyone was really ready to go for it and we calmed that down.
"The first goal was a great goal and the second goal was outstanding - we would have loved to have ended it earlier but we take the three points and keep going."
However, there was bad news for Liverpool, with Klopp's post-match confirmation that centre-back Joel Matip has ruptured an anterior cruciate knee ligament - an injury sustained during Sunday's win over Fulham.
-- Courtesy of BBC Sport