Patrick Vieiraâ€™s double either side of half-time kicked things off for City and their manager Roberto Mancini was delighted with the showing after a tough Europa League trip to Greece during the week.
"In the first half an hour Notts County played very well," he said.
"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."