City looked set to open the campaign with five successive top-flight wins for only the second time in their history when they cruised to a 2-0 lead at Fulham.
They could easily have been further ahead but relinquished their stranglehold enough to allow the home side to come roaring back to salvage a 2-2 draw.
The result was the first blow to City's title hopes, allowing rivals Manchester United to open up a two-point lead at the top of the table.
It was also the third time in just over a week that they have not taken enough of their chances and manager Mancini said: "This could be a lesson for us.
"Because when you dominate a game like we did today, and you score only two goals, it's a problem.
"For 10 or 15 minutes after Fulham scored, they played very well.
"But a football match is 95 minutes and a top team like us needs to have more control of the game.
"I'm not angry. I'm just disappointed because we conceded two stupid goals."
Mancini's men did not need to get out of second gear in the first half, Sergio Aguero maintaining his fine start to life in England by putting them ahead.
Chances came and went before Aguero notched the eighth goal of his five-game Premier League career just 12 seconds into the second half.
But Fulham quickly pulled one back through Bobby Zamora before Danny Murphy equalised via a wicked deflection 20 minutes later.
Mancini, who withdrew attacking trio David Silva, Aguero and Samir Nasri in a bid to give his side more bite, said: "For 10 minutes, we defended with five or six players. It is not good enough."
Mancini also appeared to concede his players were not currently showing the mentality of champions.
"Probably, probably," he said, before claiming they had been tired by their midweek Champions League exertions.
"If we had other players, probably we would make other changes.
"I don't have players at the moment. I can only change the full-backs.
"I have only two midfielders, because James (Milner) and (Nigel) De Jong have injuries."
It was pointed out that Fulham also had injuries and had played in the Europa League 24 hours after City's Champions League game.
Mancini claimed the Cottagers had made "five or six changes" but in fact they had made only four, with the Italian making just two.
Home boss Martin Jol said: "We were a bit tired but you could say they were a bit tired as well.
"It was a fantastic achievement to come back against them. I am not sure if they will give away leads like this in the future.
"To come back from the death is what we probably did, but we scored and we had to believe."
Unlike Mancini, Jol was delighted with his side's conversion rate after they struggled to put away Aston Villa and Blackburn in their previous two home games.
"Today we had four opportunities and scored two goals - that is good," he said, after a result which lifted his side off the foot of the table, even if it left him still searching for his first league win in charge.
"The spirit will always be 100% if you get results, and we didn't have the results - as simple as that.
"To show against a team like that you can come back after scoring two goals must have given us a big boost and confidence for the future."
Jol admitted it had been "awful" propping up the table.
He said: "We played two away games and had to play in Europe as well so it wasn't easy, but you have to win your home games.
"Luckily enough, for now, we are 18th. It looks better."