Ireland, who won three of their five qualifiers prior to kickoff, came under increasing pressure, and the hosts ultimately succumbed to it.
First, Marco Benassi expertly volleyed home a dropping ball on 28 minutes, before Valerio Rosetti nodded home a Gaetano Monachello cross nine minutes from half time as the visitors flipped the clash on its head.
There was further misery for the Boys in Green just before the hour mark when AC Milan star Alessio Romagnoli tapped home a Danilo Cataldi cross from close range to put the Italians on the brink.
Noel King's side almost reduced the arrears on 75 minutes, but first-choice Milan goalkeeper Gianluigi Donnarumma pulled off a smart point-blank save to keep out Byrne.
It was the wake-up call Italy needed, as the visitors forced Darragh Lenihan to divert the ball into his own net with seven minutes left.
The Irish stay fourth in Group 2, four points behind leaders Italy.