It looked set to be a frustrating evening for the Azzurri, who dominated possession and spurned a glut of gilt-edged opportunities.
However, Biraghi slid at the back post to steer home Kevin Lasagna’s flick on from a corner with only seconds remaining to prompt euphoric celebrations from the visitors.
It was no less than they deserved following 18 attempts at goal, with Jorginho and Federico Chiesa both denied by the crossbar in the first half.
Poland, who will finish bottom of Group 3 and face relegation from Nations League A, had their own chances with Kamil Grosicki perhaps wasteful in front of goal.
Italy, though, started as they meant to go on, Jorginho’s first-time shot clattering the woodwork in the opening seconds.
Lorenzo Insigne’s chip sailed over the bar while Federico Bernardeschi snatched wide a little later as Roberto Mancini’s men laid siege to Poland’s goal.
Chiesa stretched to get the final touch to a tantalising ball into the area but saw his effort come back off the frame of the goal.
Jorginho was proving something of a menace in the middle of the park and he muscled his way into the area before rifling a low shot that brought an excellent save from Wojciech Szczesny.
The Poland goalkeeper was also on hand to palm away Italy captain Giorgio Chiellini’s header that bounced off the ground.
Alessandro Florenzi exchanged passes with Insigne before shooting but Szczesny once again pulled off the save to make sure the score remained level at half-time.
Italy’s misfortune showed no sign of abating after the interval although they nearly fell behind when a defence-splitting pass saw Grosicki go through on goal.
However, Gianluigi Donnarumma was equal to the Hull midfielder’s effort, with Piotr Zielinski’s follow-up sailing over.
Italy finally had the ball in the net midway through the second half when a fine passing move ended with Chiesa slotting past Szczesny, but the goal was chalked off for offside in the build-up.
Bernardeschi then ghosted into the area unchallenged but, with the goal gaping, was unable to direct his free header on target.
At the other end Grosicki was once more put clean through but, instead of taking a touch, he decided to shoot from the edge of the box and Donnarumma was able to parry.
Still Italy kept coming forward and their resolve was rewarded at the death, Biraghi getting the final touch after substitute Lasagna headed towards the back post from a corner.
Victory lifted Italy to within two points of group leaders Portugal, who have played one game fewer.