The Red Devils were second best at Old Trafford as they surrendered to their bitterest rivals.
Jurgen Klopp’s men pounced on United’s inept defending to establish an early 2-0 dominance courtesy of goals from Naby Keita and Diogo Jota before the opening 15 minutes.
In-form Mohamed Salah got his first in the game on 38 minutes before his second on the cusp of halftime made the scoreline 4-0.
The Egypt international completed his hat-trick after getting on the end of a wonderful pass from Jordan Henderson before getting past David De Gea.
Solskjaer, who is under intense pressure to turn the corner at Old Trafford, said he was the one at fault against Liverpool.
"I have heard nothing else," the Norwegian coach said. "I am still thinking about tomorrow’s work. Of course, they are all low.
“I have never felt any worse than this. This is the lowest I have been. I accept the responsibility. That is mine today and it is mine going forward.
He added: "It is the worst feeling. The feeling that we are getting to where we want to be has been there. Last few weeks we have hit a brick wall.
“We have conceded too many easy goals. That is a concern. This is the lowest I have been. I accept the responsibility. That is mine today.
"I do believe in myself and I am getting close to what we want at the club. The results lately haven’t been good enough.
“Hands up. Next week is Tottenham away. Then Atalanta. Then Man City. We have to go into this with the right frame of mind.
"The time is now. Man Utd have had difficult times before and we have always bounced back. We have started this season badly.
“We have hit a brick wall. We need to start building results – and get clean sheets."
