United were 4-0 down at halftime for the first time in the Premier League after 45 minutes of shambolic defending and the sense of chaos was added to when substitute Paul Pogba was sent off on the hour mark after entering play after the break.
The United hierarchy has shown faith in Solskjaer throughout difficult periods in his nearly three years in charge of the club but will come under pressure to rethink their loyalty to their former player after such an embarrassing performance and result.
"We know we are rock bottom, we can't feel any worse than this," said Solskjaer, who said though that he had no intention of falling on his sword.
"I have come too far, we have come too far as a group. We are too close to give up now."
Juergen Klopp's Liverpool, who are in second place, a point behind leaders Chelsea, looked every bit the title contenders they are with Salah continuing his remarkable run of scoring form -- finding the target for the 10th game in all competitions.
After their difficulties with injuries last season. Liverpool look just as strong as the team who won the title in 2020 with their attack once again among the most devastating in the game.
"What we did in the last third was insane. Pressing high, winning balls, scored wonderful goals," said Klopp.
"5-0, I couldn't be happier, it is exceptional. Absolutely outstanding," added the German.
It should have been United though who opened the scoring when, in the fourth minute, Bruno Fernandes had time and space inside the box but fired his shot wide.
Liverpool showed no such generosity a minute later when Luke Shaw sloppily played Salah onside and the Egyptian's pinpoint cross was turned in by Naby Keita.
Shaw was at fault again, eight minutes later, when he and Harry Maguire got in each other's way and Keita took advantage moving the ball outside to Trent Alexander-Arnold whose low ball across the face of the goal was slid in by Diogo Jota.
Salah then made it 3-0, his initial shot from the edge of the area was blocked but United's defence failed to react and Keita pounced on the loose ball, pulling the ball back to the Egyptian who drilled a shot high into the net.
Cristiano Ronaldo was then lucky to stay on the field after picking up only a yellow card for kicking out at Curtis Jones while the Liverpool midfielder was on the ground adjacent to the ball.
There was simply no authority or sense of discipline in the United defence and Salah made it four on the stroke of halftime with a near post conversion of a low ball in from Jota.
It was the first time United had been 4-0 behind at the interval in the Premier League era and they were loudly booed off the field.
It took just five minutes after the break for Salah to complete his hat-trick with a trademark clinical finish after Jordan Henderson won the ball from Pogba in midfield and curled a pass into the path of the forward.
United's day got worse as they had a Ronaldo goal over-ruled for offside by VAR and then substitute Pogba was sent off for a two-footed tackle on Keita, 15 minutes after coming on.
That was a sign for Liverpool to finally take their foot off the gas and keep possession while their supporters taunted Solskjaer and the United fans for the final half hour of the game.
It was a ruthless display from Liverpool, their biggest win at Old Trafford, and a calamitous one from their rivals and it remains to be seen just what the fallout from this humiliation will be.
West Ham United handed Tottenham Hotspur their fourth defeat as Michail Antonio scored a second-half goal to earn his side a 1-0 home win.
West Ham moved up to fourth with 17 points while Spurs -- who also lost to Crystal Palace, Chelsea and Arsenal -- fell to sixth as Nuno Espirito Santo's side failed to test the home side's goal in the second half.
Leicester City secured a 2-1 win at Brentford thanks to goals from Youri Tielemans and James Maddison.
The result lifted Leicester up to ninth in the standings with 14 points, while Brentford dropped to 12th on 12 points from nine games.