The hosts grabbed the lead against the run of play on 28 minutes through Bulgarian winger Martin Petrov's far post tap-in and Carlos Tevez poked home before the break to double the lead.
City, playing their first game under Italian Mancini since Mark Hughes's 18-month reign as manager ended last Saturday, remained sixth following their eighth league win of the season.
"I am very happy because we won a hard match. I say thank you to the players because they are fantastic," the 45-year-old told Sky Sports.
"I think that we must improve, but today I am very happy for the players."
Donning a blue and white City scarf, the Italian received a warm welcome from the home fans prior to kickoff although his decision to leave in-form forward Craig Bellamy on the bench did not prove popular.
"They (City fans) were fantastic. For an hour the fans must be our 12th man on the pitch," added Mancini, who won three Serie A titles with Inter Milan.
City did include Brazilian international striker Robinho in the line-up although it was the visitors who pressed first when striker Tuncay Sanli's effort was well blocked by goalkeeper Shay Given.
Mancini's decision to switch wingers Petrov and Robinho shortly after quickly paid dividends as the hosts took the lead albeit in scrappy fashion as Petrov salvaged an embarrassing Robinho mis-hit to tap home a Tevez cross.
They made it two seconds before the interval when poor defending from Stoke, now without a win in five league games, allowed an unmarked Tevez to notch his 10th of the season.
Despite the scoreline City were largely unimpressive and struggled to build momentum after the break, forcing Mancini to end Robinho's afternoon on 70 minutes after a dismal display.
The 32 million pound striker's exit was greeted with ironic applause by the Eastlands fans who increased the volume as Bellamy stepped on as his replacement and continued to impress.
Manchester City travel to Wolverhampton Wanderers on Monday before playing away at Championship side Middlesbrough in the FA Cup third round next Saturday.
