The leaders showed were susceptible at the Stadium of Light as Sunderland survived a battering before snatching victory deep into injury time.
Substitute Ji Dong-won made the most of a counter-attack as too many men in blue stood and watched, and the chance to establish a gap over rivals Manchester United evaporated.
Asked if that was good enough, Mancini replied: "No, not for a top team. In that situation, we attacked with eight players and the same players who attack need to come back and run quickly.
"I know football and for that reason in the last 15 minutes, I continued to call [central defenders] Vinny [Vincent Kompany] and [Joleon] Lescott to say, 'Stay there, stay there, don't leave the space', because that was dangerous.
"We continued to attack and sometimes we wanted to attack with 10 players. This is impossible.
"But that's it, it's finished. We have a game in 48 hours and we need to recover now."
Liverpool are next for City on Tuesday, when Mancini will expect a positive response from his players.
The omens were not promising for Sunderland when boss Martin O'Neill was forced to do without goalkeeper Keiren Westwood, defenders Phil Bardsley and Titus Bramble and midfielder Kieran Richardson.
But O'Neill's side produced a performance of real grit to give themselves the opportunity to snatch a remarkable victory at the death.
Sunderland created arguably the game's best two chances before the dramatic finish, with Nicklas Bendtner with three minutes gone and Stephane Sessegnon 19 minutes after the break failing to make the most of them.
For City, Edin Dzeko was unable to take any of the opportunities which came his way, while it took a fine save from home goalkeeper Simon Mignolet, playing in a protective mask and for the first time since fracturing his nose and an eye-socket in October, to deny substitute Sergio Aguero from point-blank range.
Micah Richards, a 67th-minute replacement for Aleksandar Kolarov, hit the bar with a minute of normal time remaining, but it was Ji who finally snatched the win in the fourth minute of stoppage time when he rounded Joe Hart from Sessegnon's pass to spark wild celebrations at the Stadium of Light.
There was a suspicion of offside about the goal, although after seeing his side concede a controversial penalty on Boxing Day, O'Neill was delighted to have received the benefit of the doubt.
O'Neill said: "They tell me it was marginally offside - marginally, marginally - and we ended up winning the game.
"It's been an incredible weekend for teams and we are delighted with three points."
Mancini, who headed down the tunnel before the final whistle, admitted his side had been made to pay for missed chances.
He said: "It's frustrating because we can't lose a game like today's. In the second half, we had I don't know how many chances to score.
"But football is this. Sometimes you can't win. Sometimes you score three or four goals in a game, sometimes you can't score even if you have 10 chances to score.
"I am disappointed that we conceded a goal because a strong team cannot concede a goal like this."
O'Neill, who insisted he had no problem with Mancini's early departure from pitchside at the end of the match, was thrilled to have secured the points.
Asked if he was delighted, he said: "What do you think? Absolutely, yes. It was just a remarkable performance by the team.
"The roar for the goal was deafening, and I was delighted to hear the roar for the final whistle as well.
"It was a fantastic effort, really, really fantastic."