The reigning Barclays Premier League champions picked up a 2-1 victory at Fulham thanks to substitute Edin Dzeko's late effort.
Had the Bosnian not found the net, though, the draw would have seen City go five matches without a win in all competitions for the first time in nearly four years.
Mancini admits his team are yet to find the form they showed last season but believes they are moving in the right direction.
"I was worried because we continued to play well and had chances," Mancini said.
"We didn't score for 87 minutes and I am very pleased for Edin because he scored an important goal.
"It is also difficult for me to choose 11 players because I have 17, 18, 20 players that are really good.
"We play every three days so we need all of these players. Fulham is a difficult pitch so it was really important to have all these good players.
"I think that we were very close to the same we played last year.
"The problem we had was we didn't have all the players in pre-season. Twelve players started their pre-season 10 days before the championship began, which is difficult.
"I think, in my opinion, maybe after the international break we will be OK."
Saturday's win stopped the rot and propelled City up to fourth, keeping them within touching distance of Manchester United and Chelsea.
"We were unlucky for three or four weeks and we deserved to win," Mancini said.
"Against Fulham we had a lot of possession, a lot more chances to score and I think we played very well.
"I am happy with that and if we had lost you could say it was a bad moment, but we wanted this victory and we got it."
Mancini's attention now turns to the Champions League tie with Borussia Dortmund on Wednesday for which the Italian says he has no new knocks after yesterday's victory.
Fulham, meanwhile, will be looking for a return to winning ways when they travel to Southampton next weekend.
The Cottagers have enjoyed a bright start to the season and began well against City, with Mladen Petric netting from the penalty spot 10 minutes in.
However, Sergio Aguero's leveller and Dzeko's late goal saw Martin Jol's side lose out on a point the Dutchman believes would have been fortuitous.
"Everybody saw before the game they changed their team," Jol said.
"Normally they play with Aguero and striker Dzeko, but they left him out and played 4-3-3.
"They don't do that all the time and that was good for us because they showed us respect.
"Scoring the goal after 10 minutes gave us a good cushion but we need a bit more to score the second one.
"To score the winner at 1-1 we had one good chance through Bryan Ruiz and Alex Kacaniklic, but it was just not enough for me to say we probably deserved the point.
"On the other hand, I feel if you work that hard and concede a goal just before the end through our own mistake it's a bit frustrating."