"This is the end of Ole Gunnar Solskjaer. Thank you Liverpool," one fan wrote on Twitter
"If Solskjær isn't sacked tomorrow I'm done," wrote another.
Solskjaer is yet to win a trophy since taking charge in December 2018 but has been credited with leading the club back into the Champions League in the past two seasons.
And he insisted he will not step down.
"I have come too far, we have come too far as a group. We are too close to give up now," he said.
"The players will be low but there's loads of characters there.
"We know we are rock bottom, we can't feel any worse than this. Let's see where we take it."
Meanwhile, Liverpool's Mohamed Salah, who wrapped up his hat-trick just after the break, was elevated by the win speaking to Sky Sports after the match.
"We knew they will come and try to give everything so we had to give 100 per cent," he said. "We just tried to play our football and play in between the lines to create more chances. We did that and were lucky to score.
"We know what it takes to win the Premier League, we did it two years ago. A big win here but still only three points.
"We try to win each game and try to win the title. We need to play to win everything. From our heads from pre-season, we need to win the Premier League and hopefully we go for that."
Liverpool's first victory in front of a crowd at Old Trafford under Jurgen Klopp moves them back above Manchester City into second, just a point behind leaders Chelsea.