Mancini was indebted to two brilliant saves from Shay Given, the first to deny Tuncay when the game was still scoreless, the second from a ferocious James Beattie effort as Stoke were enjoying one of their better spells.
It meant first-half goals from Martin Petrov, his first in two months, and Carlos Tevez, who finished neatly after Gareth Barry had cushioned a header into his path, were enough to secure maximum points at the end of a turbulent week.
For Mancini it was the perfect ending, having been delighted to hear the applause ringing around Eastlands when he made his introduction just before kick-off.
"The atmosphere was brilliant," said Mancini, who didn't do his popularity any harm by wearing a blue and white City club scarf that he still had not removed 45 minutes after the final whistle.
"It was a great experience. And I was happy the fans gave me a good reception.
"Mark Hughes was very popular. I am not popular here, just in Italy. But I can answer the fans with my work."
Stoke boss Tony Pulis has had his own Christmas troubles following the well-publicised spat with Beattie, who again started on the bench and seems certain to leave at some point during the transfer window.
"The first goal was going to be important today," he reflected.
"If we had scored first it would have been interesting.
"Had Tuncay's chance gone in, the reaction would have been interesting, not just from the City players but the supporters as well."