Also on Thursday, Kosovo stunned Slovakia 4-3 away to set up a clash against Turkey with Tuesday's winner joining the Socceroos in Group D at the World Cup.
The Italians, managed by the inspirational Gennaro Gattuso, had to thank Sandro Tonali and Moise Kean for second-half goals.
The breakthrough arrived in the 56th minute when Tonali pounced on a cleared cross, letting the ball bounce once before firing into the bottom corner from the edge of the area.
Kean effectively ended the contest in the 80th minute, latching onto a Tonali pass and ghosting past his marker before dispatching a low strike into the net.
"I felt the whole country was on my shoulders," Kean told Sky Sports about his goal. "There is one more game to go, we need one more step, but we are feeling positive and need to keep going."
The four-time world champions will face Bosnia and Herzegovina on Tuesday for a place at the tournament in the US, Canada and Mexico in June-July.
In Bratislava, Kosovo twice came from behind to beat Slovakia 4-3 away in a thrilling playoff semi-final and earn the right to host Turkey at home in Pristina.
The Balkan nation only began playing international football since 2014 and have never qualified for a major tournament, but are now one game away from joining Australia, Paraguay and the United States.
Slovakia took a sixth-minute lead when fullback Martin Valjent headed home from a corner, but Kosovo equalised with their first effort on goal via a smart finish on the turn by Veldin Hodza.
Slovakia were back ahead on the stroke of halftime as Lukas Haraslin's free kick from out on the left wing evaded everyone to creep into the goal.
It took all of 60 seconds into the second half for Kosovo to pull level again with Fisnik Asllani before Florent Muslija's free kick crept in at the near post to see them go ahead 3-2 on the hour mark.
Kreshnik Hajrizi crashed home Kosovo's fourth from point blank range, snapping up a loose ball in the 72nd minute and they kept the hosts at bay until four minutes into stoppage time when David Strelec handed Slovakia a brief glimmer of hope before the final whistle confirmed their elimination.
Turkey won the other European playoff semi-final earlier with a 1-0 home win over Romania in Istanbul.
They took the lead shortly after the break when Ferdi Kadioglu was set up in front of the goal with a long, accurate pass from Arda Guler, the 21-year-old winger for Real Madrid who could become one of the World Cup's younger standouts.
Vincenzo Montella's Turkey squad also reached the quarter-finals of the 2024 European Championship.
In other results, Sweden beat the Ukraine 3-1, Poland beat Albania 2-1 and Denmark beat North Macedonia 4-0.
Bosnia and the Czech Republic needed penalty shootouts to eliminate Wales and Ireland respectively.
TUESDAY'S PLAYOFF MATCHES:
PATH A:
Bosnia and Herzegovina v Italy in Zenica
PATH B:
Sweden v Poland in Solna
PATH C:
Kosovo v Turkey in Pristina
PATH D:
Czech Republic v Denmark in Prague
with agencies
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