Ole Gunnar Solskjaer's side suffered the biggest defeat to its fierce rival since 1895 as it lost a league match at Old Trafford by five or more goals without scoring for the first time in 66 years.
Naby Keita, Diogo Jota and two goals from Mohamed Salah meant United trailed by four goals at half-time of a Premier League game for the first time, before Liverpool's Egyptian star completed his hat-trick in the second period – the first treble scored by a visiting player at Old Trafford since Ronaldo for Real Madrid in 2003.
Paul Pogba was sent off just 15 minutes after being introduced as a substitute to compound a truly miserable day for United fans, some of whom booed their side off at half-time and did not stay for the second period.
Solskjaer described it as his "darkest day" as United manager and there is now significant pressure on the shoulders of the Norwegian, who has overseen five defeats and just three wins in the last nine matches.
Shaw, United's players' player of the year last season, is one of several players to have endured a poor run of form and the England left-back believes now is the time for everyone at the club to reflect and take action.
"We are extremely disappointed, it is not good enough, and it hurts a lot," he told MUTV. "I think football is obviously a team sport, we are in it together and we are all together. But I think, as individuals, we need to take responsibility for some of the actions tonight.
"Personally, that is of course why I am here, I am not hiding inside. I know I was not at my best tonight, I know I was not good enough, and of course I will take some responsibility for that.
"It is something that we shouldn't allow. We are at one of the best clubs in the world, one of the biggest clubs in the world, playing at our home stadium in front of all these fans. It shouldn't happen.
"We should be much better in these games and especially in key moments. We had a big chance in the first minute or so. If that goes in, maybe we could be looking at a different game, but that is not to blame anyone. I am not here to blame individuals. It is a team game.
"I just want to thank the fans because 5-0 down is never easy for them and, of course, it is not easy for us, but they stuck with us, they carried on singing and they could have easily left the stadium. We would have totally understood that with the way we performed and the scoreline, but they stuck with us, they kept on singing and we really appreciate that. I can only apologise for the performance.
"I know we can better, I know we have to be better. I think it is not just us as players. The whole club needs to have a look at ourselves to see what went wrong and what is going wrong. We need to reflect on that and we need to do something about it.
"It is not good enough for the standards this club sets. We should be challenging for leagues and for trophies. We haven't been doing that for a while now. It is not what this club expects. We all want to be doing that but with performances like that there is no way we can be doing that. We have to be honest, we have to be honest with ourselves in the changing room, speak about it."
Shaw felt the Liverpool defeat had been on the cards given the standards of United's recent performances, saying his side has become far too easy to play against.
"It's ourselves we need to look at first and foremost in the mirror. Are we doing everything right and preparing right for the games in ourselves?" Shaw said. "We have the tactics and how the manager wants us to play.
"At times, we're too easy to play through against. You look at the first goal, it can't be possible that they can have three running through in the first five minutes. We need to be more compact, we need to be better.
"This result was coming. In past games where we've won, we haven't been at our best. We felt that inside the dressing room and today we have to reflect and we have to move on from this because it hurts.
"I think we all know when we're at our best we're capable of playing well. It's not just a team, it's individuals who are not at their best. Me, myself, I'm not. I need to look at myself, see what I'm doing wrong and focus on that.
"At times in the game, we're far too easy to play through against, not just today but it's been a number of games now where it's been happening bit by bit and we can feel that in each game.
"It happened today and, against a great team like Liverpool, they're going to punish us and they did."