City cruised past the Championship side to progress to the quarterfinals, with Sergio Aguero scoring goals on either side of Tevez's strike after Yaya Toure had opened the scoring.
Mancini's side has slipped behind in the Premier League title race but was glad to take the positives from the FA Cup win and believes it could help the likes of Tevez return to top form.
"Carlos is an important player for us, also [even when he] is not in good form," he told ESPN. "He's experienced, he fights and he didn't play for six days; maybe today he played 60 minutes and it was important he scored one goal. It's good confidence for him.
"We need our strikers to score more like today. If our strikers start to score I think that we can have another chance in the Premier League."
The 2011 FA Cup winners go into the last eight of the competition as one of the big favorites to earn a place at Wembley and Mancini was aware of the importance of starting well to avoid an upset.
"I think that it was really important to win this game," Mancini said. "Clearly if you score two goals in the first 20 minutes it's easier.
"But the FA Cup is a difficult competition. Arsenal [lost to Blackburn] yesterday, Everton [drew with Oldham], and other teams that play in the Premier League are out of this competition.
"If you don't play with concentration, it's difficult."