The hosts created numerous chances but lacked edge, in a game that looked to be heading for a surprise goalless draw.
It was Gareth Barry's stoppage-time winner that saw off hard-working Reading, piling more woe on the Premier League strugglers.
Mancini said: "I think we deserved to win but when you are in the last two or three minutes of extra time, it could be difficult.
"These games, if you don't score in the first half when you have a lot of chances to score, I think they can be strong.
"They played well, they defended well, for 90 minutes with all players behind the ball. We didn't find space to score.
"When you have one chance, you have to score. In the end, I am happy with the victory but we had a big problem."
The match was City's last home game of 2012 and they won it in a manner in keeping with the rest of the year, notably last season's title decider.
Mancini added: "In this we have improved a lot. We want to recover so when we are losing we try until the end to score, to recover the game.
"Our character is strong. Sometimes you can win, sometimes you can't but I am very happy because we have recovered a lot of games like today.
"But it is more important to score before. If not, my heart is not strong."