Midway through the first half City took control and in the 37th minute Patrick Vieira nodded in the opener.
His second header 12 minutes after the interval put the hosts in the comfort zone and the scoreline was given something of a distorted look with goals from Carlos Tevez, Edin Dzeko and Micah Richards in the last six minutes.
"In the first half an hour Notts County played very well," admitted Mancini, who felt the club's Europa League trip to Greece on Tuesday affected their performance.
"It was an important win. I think for us it (the FA Cup) is important. We started the game so slowly because when you play every three days it is difficult to play 100% in every game.
"But after we scored it changed the game. We had more chances in the second half and it was important we scored those goals."