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.

Positives for Wilder despite defeat

If the Bramall Lane board were after an immediate reaction by reappointing self-confessed Blades fan Wilder, they got it.

"He's one of our own," chanted the home supporters prior to the first whistle and Wilder's demands for a return to the Blades' core values, after a number of embarrassing defeats, were certainly met.

A team who had conceded 39 goals in their previous 14 league games - including eight to Newcastle and five to both Burnley and Arsenal - gave free-scoring Liverpool relatively few sniffs in open play.

Indeed, it was the Blades who fashioned the best chance of the opening half-hour as Cameron Archer robbed Joe Gomez and fed James McAtee, whose shot was well saved by Caoimhin Kelleher.

Chief among the positives for Wilder will have been the efforts of Archer, a real livewire on the counter, who caused the Liverpool defence constant concern but just lacked support at the vital times.

Brentford and Luton are the next two visitors to Bramall Lane and, despite only taking five points from a possible 45, United are far from cast adrift thanks to the paucity of points gathered by their rivals at the foot of the table.

"First and foremost I asked for a performance and I definitely got that but I'm disappointed we haven't got a result," Wilder told BBC Match of the Day.

On his welcome back, he added: "It will stay with me forever, that. I enjoyed watching a Sheffield United performance tonight and the reception they gave me was incredible."

Ruthless Reds get the job done

While this was from a classic Liverpool display, boss Jurgen Klopp will have taken delight in the businesslike manner they handled proceedings, considering the occasion and partisan atmosphere.

Stand-in goalkeeper Kelleher - arguably at fault for two of Fulham's goals at Anfield on Sunday - made that key early save to thwart McAtee when the Reds were caught on the counter.

And Alexander-Arnold continued to answer his critics with yet another goal involvement, providing the delivery for Van Dijk's opener.

He now has 75 assists in all competitions for Liverpool - only Kevin De Bruyne (129) and Mohamed Salah (80) have more in the Premier League since Alexander-Arnold made his debut in 2016.

The biggest negative for Klopp was the sight of Alexis Mac Allister limping off to add to an injury list that includes first-choice keeper Alisson, Diogo Jota, Joel Matip, Andrew Robertson and Thiago.

Salah was denied his 200th Liverpool goal by a fine save from Wes Foderingham, who also excellently foiled Nunez one-on-one, but the Reds still claimed a first away league win in five attempts.

There was still the odd sign of the defensive issues that have plagued Liverpool this season but they registered a first clean sheet in nine away trips.

However, they did enjoy the benefit of officiating decisions that could have gone either way for both of their goals as well as over a possible nudge, which went unpunished, by Ibrahima Konate on McAtee in the Liverpool area.