The Dutch centre-backfired home from a first-half corner and Szoboszlai added a second at the death to give the Reds all three points in South Yorkshire to extend their Premier League unbeaten run to eight games.
Jurgen Klopp’s men closed the gap on league leaders Arsenal to just two points.
Wilder was condemned to defeat in his first game back as Blades manager, two-and-a-half years after he left in March 2021, following Tuesday’s sacking of Paul Heckingbottom.
This was United’s 12th defeat from 15 games and although they are only four points behind Luton, they look to have a mountain to climb.
The 56-year-old was given a warm welcome before the game and his side started with enthusiasm, creating two openings in the first 20 minutes.
It was Archer who had the next chance as he was given space to run in on goal until Van Dijk came across and blocked his shot at the vital moment.
Liverpool were far from their fluent best but began to take control and took the lead in the 37th minute.
Van Dijk was left completely free from Trent Alexander-Arnold’s corner and he stroked home from 12 yards for his first goal of the season.
Foderingham made an even better save 10 minutes into the second half as he showed brilliant reactions to palm over Mohamed Salah’s volley from another Alexander-Arnold corner.
The Blades managed to get a foothold in the game but their lack of quality in attack ensured that Kelleher was untested in the Liverpool goal as he filled in for the injured Allison Becker.
It was the Reds who had chances to kill the game as an incisive pass from the creative Alexander-Arnold played in Darwin Nunez but Foderingham again came to the Blades’ rescue.
United threw bodies forward in the final 10 minutes in a bid to try and earn a point, but they again failed to trouble Kelleher, with McAtee’s early effort their only shot on target in the match.
Instead, it was Liverpool who found a late goal, Szoboszlai clinically firing home after Nunez had won the ball back deep into time added on.