Sergio Aguero wasted three clear-cut chances before Pablo Zabaleta finally put the home side ahead with a finish from close range in the 42nd minute.
Edin Dzeko doubled the Citizens’ lead when Asmir Begović could only parry Aguero’s effort into the path of the Bosnian, who was left with a simple tap-in.
Mancini’s men then cemented their advantage in 72nd minute, with Aguero netting from the penalty spot after Steven N’Zonzi had fouled David Silva.
“We played very well, like we played against Norwich and Sunderland, but sometimes it’s not enough to play very well, it’s important to score” Mancini told BBC Sport.
“In the first half we deserved to score two or three goals, but it was difficult because Stoke is the best defence in the Premier League.
“It was important to keep a clean sheet, because if we start to score and don’t concede a goal we can have a chance to recover these points.”
Tony Pulis refused to criticise any of his Stoke players despite the defeat, with the Potters dropping to ninth in the Premier League table.
“We’ve come here and played right at our maximum, and they’ve played below par. We didn’t play as well as we can play, and they played well,” Pulis said.
“Sometimes you have to be a little bit grounded, not get carried away and look at facts and figures. When you have spent that much money on the quality they have, it’s always difficult.
“We’ve played Tottenham, Liverpool then Southampton with 10 men for the last 25 minutes so it’s been tough.
“Coming here, against one of the best teams in the Premier League, the lads looked tired in the second half. They kept going, they had great spirit, great commitment and I’m not criticising them at all.”
MORE: Latest Man Utd news
MORE: Latest Arsenal news
MORE: The latest football news