United, who conceded four goals during a shocking first-half performance, suffered their heaviest-ever home defeat to rivals Liverpool and their largest defeat at Old Trafford without scoring in 66 years.
"It is the darkest day I have had leading these players," Solskjaer told Sky Sports. "We were not good enough individually and as a team. The whole performance was not good enough.
"I choose the way we approach the game, we were not clinical enough and gave too many spaces and when you give good players space they score."
Solskjaer said the result was "miles worse" than the 6-1 home defeat to Tottenham Hotspur last season and comes just a week after a demoralising 4-2 defeat to Leicester City at the King Power Stadium.
The 48-year-old, who signed a new contract in the summer, is approaching his third anniversary as United boss and insists he is still the right man for the job.
"I've just got to say we have to get over this as quickly as we can," he said. "I have come too far, we have come too far as a group. We are too close to give up now.
"It is going to be a difficult one. 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."
United, who had Paul Pogba sent off just 15 minutes after coming on as a half-time substitute, have now slipped to seventh in the table and face Tottenham and Manchester City in their next two league games.
Afterwards, captain Harry Maguire accepted the result was "embarrassing" before sending an apology to supporters.
"I look at those lads each day in training and we are working hard as we can, but at the moment, it is not good enough," he said. "I apologise to the fans, we have to look at ourselves as individuals but also stick together.
"We apologise to the fans, it was nowhere near good enough for this club. They stuck with us right to the end and we appreciate that but as a club we have to do better.
"I am so disappointed, we gave them so many chances. They were so clinical, we had a lot of chances ourselves, but I am a defender and to concede four in the first half especially at Old Trafford is nowhere near good enough."