City remain unbeaten in the league, but missed the chance to leapfrog back over rivals Manchester United, who beat QPR on Saturday, when they failed to find a way to break down Chelsea, playing their first match under new interim manager Rafael Benitez.
There were few real chances of note in a match played out in an often tense atmosphere as home fans vented their frustrations on the appointment of the former Liverpool boss to replace the popular Roberto Di Matteo.
City were the more creative, with David Silva heading Pablo Zabaleta's cross over, before Sergio Aguero nodded straight at Petr Cech just before half-time.
After that, there was little to set the pulses racing from the champions, before Matija Nastasic powered a late header into Cech's arms from a corner.
Mancini knows City must again find their cutting edge in attack, with a midweek trip to Wigan next before hosting fifth-placed Everton.
"I am not satisfied with one point. To play here at Chelsea is always difficult, but to play well as we did here, we can't lose two points," he said.
"We were so poor in the last 20 metres, when we have a chance, we need to score. We missed the last pass - you have to be strong with that in the box. If you are soft, you won't score. That is why I was upset."
Mancini continued: "We are a good team and we won the last championship, we want to defend this.
"But I think that us, United and Chelsea will all fight for this title."
City skipper Vincent Kompany produced a composed display at the back, shackling the threat of Fernando Torres.
The Belgian, who had battled through an ankle problem sustained against Real Madrid, feels Mancini's side can build on Sunday's efforts, when they restricted Chelsea to just one shot on target.
"In the end it was a draw and as a defender I am happy with a clean sheet," he said.
"This is another big game out of the way for us and we look forward to the other games. We feel that we can do enough to win a lot of games from here until the end of the season."
Mancini, meanwhile, empathised with Benitez's situation.
"I don't think that Rafa is a magician, he cannot change everything in a few days," said Mancini.
"Weeks ago, Chelsea were on top and playing well. They have good players. Every manager needs time."
Benitez's fellow former Inter Milan boss added: "To do this job is very difficult. To do it is fantastic. We know everything, but it's a difficult job.
"Also, in England, it's easier than in Italy to do this thing."