The Red Devils trailed 4-0 at half-time as Liverpool ran riot, with Mohamed Salah finishing with a hat-trick, while substitute Paul Pogba was sent off on the hour.
The result means United have collected just one point from their past four Premier League games, dropping them down to seventh.
Solskjaer admitted it was a dark day for United, particularly at Old Trafford given the rivalry and context.
“It’s not easy to say something apart from it’s the darkest day I’ve had leading these players and we weren’t good enough, individually and as a team,” Solskjaer said. “You can’t give a team like Liverpool those chances, unfortunately we did.
“The whole performance wasn’t good enough, that’s for sure. We’ve created openings, they’ve had chances and they’ve been clinical. We didn’t take our chances early on and the third goal is the one that decided the game.”
Solskjaer added that he was devastated with the result, but insisted they would bounce back.
“You know, you can look back at last season and we lose a big one to Spurs, 6-1, but this is worse,” he said.
“The opposition, as well, make it much worse for me as a Manchester lad. And it’s I’ve just got to say we have to really get over this as quickly as we can and move on.
“You’re not a robot, of course. I’m devastated and I’m as low as I’ve ever been when I’ve been at this fantastic football club. We’ve had dark days here before and we’ve always bounced back.”
Liverpool, meanwhile, are one point behind Chelsea at the summit, and remain the only unbeaten team.
The Reds also maintained their stunning record of scoring three or more goals in every away game this season.
Salah also extended his stunning record to 10 consecutive games where he has scored.
Liverpool boss Jurgen Klopp was asked if the result represented one of the greatest days of his career and answered: “Greatest days? I didn’t think about it, yet. It’s a good day, a really good day and I don’t want to be disrespectful. I have no idea, but it’s a big one.”