United are now level on points with their table-topping rivals, who boast a six-stronger goal difference, but Mancini remains confident about his charges' title chances.
The Italian said: "I made mistakes, and it is an admission from me that I got it wrong - sometimes it is possible to make mistakes.
"We didn't prepare properly for this game. Maybe I thought it would be easier against Everton than it was. I should have known it is never easy coming here.
"We played well against Tottenham and won three games in the Premier League. Everton were without three or four important players, and maybe we didn't prepare properly, which was my mistake.
"But defeat is not important, because we are still top, and there are 15 games remaining, so we have time (to win the title). All the players played well, I have no complaints with them, and I think I have to learn from my mistake. I will not sleep tonight!"
Mancini was also asked about the future of Carlos Tevez, after the Argentine failed to seal a move away from Eastlands before the close of the transfer window.
He said: "I don't know his (Tevez) condition, so I don't know at this moment if he can rejoin the squad, and I don't know if he will have to apologise. At this moment I am just thinking about my mistake."