The champions looked to be in control following Carlos Tevez's first-half opener, but goals from substitutes Rickie Lambert and Steven Davis in the second half gave the home fans a fright before Edin Dzeko and Samir Nasri earned the points.
City missed a number of chances to kill the game off earlier, most notably when David Silva saw a tame penalty saved by Kelvin Davis, and while Mancini was happy with the result, he insisted his side must look to improve in the coming weeks as their title defence gets underway in earnest.
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"It's important to show character but it's better to close the game before, we had so many chances and when you don't close the game it can hurt you."
City earned crucial points last season on the back of some memorable come-backs, most notably in their 3-3 draw at home to Sunderland and their dramatic 3-2 win against Queens Park Rangers on the final day of the season.
And after watching his side come from behind to defeat Chelsea in the Community Shield a week ago, Mancini stressed that he hopes to avoid too many such matches in the coming campaign.
"It's not normal what happened last year, this [result] happened two or three times," he said. "I think today because it's the first game, we only worked for two weeks together, but we need to improve."
City's victory was tarnished by an injury to Sergio Aguero, who had to be stretchered off with a knee injury after just eight minutes and is set to undergo an MRI scan to ascertain the extent of the damage.
Mancini had no update on the striker's problem, stating simply: "We need to wait two days for this."