City produced a strong performance to beat Leeds 4-0, with Sergio Aguero scoring twice and further goals from Carlos Tevez and Yaya Toure securing a passage to the competition's last eight.
Mancini felt the result was reward for his team's concentration during a comfortable win over Leeds that the Italian hopes will also help inspire his players to break down Manchester United's 12-point advantage in the race for the league championship.
"It was really important to win this game and if you score two goals in the first 20 minutes it is clearly easier," he said.
"The FA Cup is a difficult competition and a lot of Premier League teams have gone out of it. If you don't play with concentration it's difficult.
"If he (Aguero) scores a lot of goals from now until the end, the championship is not finished."
Leeds manager Neil Warnock was not happy with City's second goal, an Aguero penalty awarded by referee Mark Clattenburg after he had adjudged Tom Lees to have fouled the Argentine striker in the box.
Contact between the two looked minimal, but Aguero stopped playing after Lees' arm made contact with his face.
Mancini, though, insisted he did not have a good view of the incident but that the result was the right one.
"It (the foul for the penalty) was difficult to see from the bench. I didn't see it," he said. "The game finished 4-0 and I don't think we gave them many chances to score.
"I think the result is correct. I don't know if the penalty was correct, I will see it again but I can't say at the moment."