"Now we understand that if we want to win the FA Cup, we have to play at 100%. When you play against a team in the Championship, it is never easy."
The Foxes equalised through Alex King after an error from keeper Joe Hart.
"Leicester played very well," added Mancini. "They did not deserve to lose this game.
"Joe made a mistake but this can happen. He saved us in the first half and he has saved us in a lot of games."
Leicester had taken the lead at the Walkers Stadium inside a minute when Sol Bamba struck from six yards on his debut for the Foxes.
James Milner equalised and Carlos Tevez put his side in front on the stroke of half-time but the hosts pressurised their Premier League opponents after the break.
"We conceded a goal after just one minute, we slept for two or three minutes," said the Manchester City boss.
"But we improved during the game and scored two good goals. It is important we have another chance."
I am not happy about that - we should have dealt with Leicester City
New Manchester City signing Edin Dzeko, signed from German side Wolfsburg on Friday for a fee of £27m, watched the game from the stands.
"Dzeko will be an important player, he is very different from the other strikers we have here," said Mancini.
"He will be important for the rest of the season. He will play next weekend and be available against Wolverhampton."
Leicester were the better side for periods of the match and it took a goal-line clearance from Milner to prevent Bamba from scoring a third after King had equalised.
Bamba also forced a brilliant reaction save from Hart in the first half as the Ivory Coast international defender enjoyed a memorable debut.
"Sol Bamba is very dangerous at set-pieces - I knew that from when I was in charge of the Ivory Coast for the World Cup," said Leicester boss Sven-Goran Eriksson.
"He scored in a practice match and that was when I saw how good he can be.
"I hope he continues to perform like that for the rest of the season."
Eriksson, who managed Manchester City during the 2007-08 season, paid tribute to the way his Championship side competed against top-flight opposition.
"I must be pleased with the performance and it was a good football match," said Eriksson, whose team are 12th in the Championship.
"I'm very proud of my players and they should be proud of their display.
"We stood up to Manchester City, one of the best teams in the country. All the players were brilliant.
"You could not see much difference between the teams although, of course, it will not be easy up there for the replay."
Source: BBC Sport
Source: BBC Sport