Mo Salah scored a hat-trick as Jurgen Klopp's side enjoyed their biggest ever win at Old Trafford, which started emptying even before half-time substitute Paul Pogba was sent off for a foul on Naby Keita.
Former United defender Gary Neville labelled it a "monstrous day" but said that the board were "unequivocal" in their support of Solskjaer, who won the treble with the club in 1999.
Asked if he believes he still has that backing from above, Solskjaer said: "Well, I've heard nothing else (on my future) and I'm still thinking about tomorrow's work.
"Of course, we're all low. I can't say now that I've felt any worse than this. This is the worst I've been, the lowest I've been but, as I've said, I accept the responsibility and that is mine today and it's mine going forward.
"Today's rock bottom, low in confidence but next week it's Tottenham away, it's a Champions League game at Atalanta, the next team who visits us... that's Man City.
"We have to look forward, we have to sort the frames of mind out and make sure we go into the next training session and the next game with the right frame of mind."
United have now lost three of their last four Premier League games and had to come back from 2-0 down to beat Atalanta in the Champions League last week.
Neville, who was working as a pundit for Sky Sports, said after the match: "I didn't think it would ever get as bad as that today.
"That was an absolute shocker, and the timing couldn't have been worse because it has been building now for a few weeks, against what I've called half-decent teams.
"As soon as they've played a proper team they've been obliterated, pulled to pieces, dismantled."
Liverpool manager Jurgen Klopp was delighted with his side's finishing although the win did come at a cost in the shape of injuries to Keita and James Milner.
"In front of the goal, in the last third, we were exceptional," Klopp said. "Clinical and ruthless, the high press was outstanding, formation top, all these things you want to see as a coach.
"The difference is you don't usually score in each situation and we did."