European champions Portugal lead the three-team group with six points from two games with Italy on four points after three games with Poland relegated to League B with one point. PHOTO: AFP
Defender Biraghi, 26, slid in the winner two minutes into injury time for his first goal for Italy in a game his team had dominated.
It gives Mancini his first competitive victory since taking over as Italy coach last May, after the four-time world champions failure to qualify for the World Cup.
"We dominated the game, we had to score the first goal, it would have been unjust to finish 0-0," said Mancini. "All the lads are trying to do their best. It's a good win in a very good game."
European champions Portugal lead the three-team group with six points from two games with Italy on four points after three games with Poland relegated to League B with one point.
Mancini stuck with the same eleven he started with in last Wednesday's 1-1 draw against the Ukraine with Federico Bernardeschi, Lorenzo Insigne and Federico Chiesa leading the attack.
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Nicolo Barella, 21, was chosen to start over Lorenzo Pellegrini in midfield after making his international debut in Genoa.
Poland opted for the strike partnership of Robert Lewandowski and Arkadiusz Milik over Genoa forward Krzysztof Piatek, who has scored 14 goals in ten competitive games for club and country.
Italy had the best of the first half but once again demonstrated their difficulty finishing off.
Jorginho rattled the bar within the first minute, with Federico Chiesa also hitting the woodwork after half an hour in the Silesian Stadium.
Poland could thank Wojciech Szczesny for keeping them in the game with the under-pressure goalkeeper denying Alessandro Florenzi, Jorginho, Federico Bernardeschi and Juventus teammate Giorgio Chiellini in the first half.
Italy had a goal disallowed after 65 minutes following a well-worked pass between Bernardeschi and Marco Verratti with Insigne offside in the box.
Poland also had a couple of chances in the second half with Kamil Grosicki denied by Gianluigi Donnarumma, with Arkadiusz Milik curling over.
Kamil Glik fired in low from inside the box following a free-kick but Donnarumma proved solid.
Just when it looked as if both sides would settle for a draw, off the last corner, substitute Kevin Lasagna, who made his international debut, steered in for the Biraghi to finish off.
The Fiorentina defender made the sign '13' with his hands, as he dedicated his first goal to former Italy and Fiorentina captain Davide Astori who died of a heart attack last March.
The result ensures Italy stayed in the tier ahead of December's Euro 2020 qualifying draw, with Poland dropping to League B.
Italy next host Portugal on November 17 with Poland playing in Portugal three days later.
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