The visitors raced to a four-goal lead in the first half thanks to two Mohamed Salah goals and strikes from Naby Keita and Diogo Jota.
There were furious scenes among the home fans, who turned on Solskjaer as he and his players trudged towards the tunnel.
It wasn't about to get any better for them after the break as Salah completed his hat-trick early in the second half when he poked the ball beyond David de Gea.
The match also saw Paul Pogba receive a straight red card for a high tackle on Keita, who was subsequently stretchered off the pitch.
Meanwhile, many Red Devils fans opted to leave the stadium early, seeing enough from the clash that will surely raise more questions about Solskjaer's future.
Liverpool were very professional but were helped hugely by some poor defending from their opponents.
The deadlock was broken in just the fifth minute as Salah played a simple pass to Keita, who was in acres of space. The midfielder made easy work of the finish as he stroked the ball in.
They nearly had their second moments later with an effort from Roberto Firmino but they didn't have to wait much longer as Jota scored in the 13th minute when he latched onto Trent Alexander-Arnold's cross as Liverpool capitalised on some indecisive defending from Harry Maguire and Luke Shaw.
Man United offered a bit going forward with long range efforts from Shaw and Mason Greenwood but their defending continued to let them down.
Keita turned provider for Salah this time as the Egyptian found the roof of the net on 38 minutes, before beating De Gea at his near post on the stroke of half time.
With the contest all sewn up, Liverpool eased their intensity but not until after Salah made it 5-0, which sparked contrasting emotions from Sir Alex Ferguson and Kenny Dalglish in the stands.
It also sparked a number of the home fans taking their leave so they wouldn't have seen Pogba's sending off, which came just 15 minutes after he was substituted onto the pitch.
Cristiano Ronaldo curled the ball past Alisson but the strike was later ruled out for offside in a match where everything that could've gone wrong for Man United, did.