Second-half goals from Bernardo Silva and Leroy Sane, with David de Gea arguably at fault for both strikes, earned City a 2-0 win at Old Trafford that sent them back to the Premier League's summit.
In an appearance as a pundit on Sky Sports, Keane was highly critical of United's performance, accusing midfielder Fred of "cheating" and claiming players "threw Jose Mourinho under the bus".
Solskjaer has indicated he will embark on a long-needed clearout of United's squad at the end of the season, with the initial new-manager bounce he provided having faded fast.
And Solskjaer accepted former treble-winning team-mate Keane may have a point about the problems at United, who have now lost each of their past three
"We played together for many, many years and he's a Man United guy, as well, he hurts just as much as anyone else connected to us, that we are so far behind City as we are.
"And that's my job, to make sure when we come back for pre-season that I have the right characters in and around [the squad]. Because that should never be allowed.
"Because when you're at Man United the spotlight will be on you, headlines, and you've got to make sure you come into work every single day with the effort like they gave today. That's your duty."
Striker Marcus Rashford criticised United's mentality after their derby defeat and Solskjaer said the players have to learn to handle the pressure of being at the club.
"That's what you sign up for when you play for Man United – the expectations," the Norwegian added. "Of course, the history, you can look back on that.
"But we are where we are now and as I've spoken to you [the media] about in the last few weeks, we have to take it step by step and make sure we're on the right track, and do it quickly.
"Because you can't linger along for many, many years, because the Premier League is so competitive and you'll suddenly be caught up by other teams."
Defeat left United in