Debutant Michael Kayode, of Brentford, scored as Italy responded well to a 2-0 home defeat to Belgium in Mancini's first match back.
The coach had led Italy to success in the European Championships in 2021, but quit after they failed to reach the 2022 World Cup. His return follows the Azzurri missing out on the 2026 finals too, the third World Cup in succession the four-time winners have failed to qualify for.
Inter Milan's Alessandro Bastoni put Italy ahead in Bursa on Monday (Tuesday AEST) after nine minutes, and Lazio's Davide Frattesi doubled the lead in the 22nd minute.
Kayode's goal gave them a commanding advantage after 27 minutes, and despite Baris Alper Yilmaz pulling a goal back for the home side early in the second period Pio Esposito quickly restored the three-goal cushion.
Esposito was paying alongside his older brother Sebastiano, of Sassuolo, who in making his debut became the third Esposito sibling to play for Italy, with Salvatore, the eldest brother, having won a cap in 2022.
"The team played well, pressed high, kept possession and played the ball through," said Mancini, whose team go to Paris on Friday to face France.
"We had to focus on this match, which was a difficult one.''
In Group A1's other match, World Cup semi-finalists France maintained their 100 per cent record under new coach Zinedine Zidane with a 1-0 win over Belgium in Brussels.
Without the injured Kylian Mbappe they left it late, Bayern Munich's Michael Olise scoring in the 88th minute from a Rayan Cherki pass.
The pair had both come off the bench 11 minutes earlier.
France hit the woodwork twice and brought five saves from Manchester United goalkeeper Senne Lammens as they peppered Belgium with 18 shots in the match.
In the competition's second tier, Sweden took control of Group B2 with a second win, goals from Celtic's Ben Nygren, Brighton's Yasin Ayari and Arsenal's Victor Gyokeres sealing a 3-1 victory over Poland.
Romania are bottom after a second defeat, beaten 4-2 at home by Bosnia and Herzegovina.
There were no such goal-fests in Group B2 with Ukraine's trip to Georgia and Northern Ireland's match at home to Hungary both ending 0-0.
In the third-tier Group C2, Montenegro made it back-to-back wins with a 3-2 victory in Armenia, while Cyprus and Latvia played out another goalless draw to earn their first points.
The Nations League, a biennial tournament featuring promotion and relegation, and a finals series for League A, also serves as part of qualification for Euro 2028.
- with Reuters
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