Matteo Pessina scored the 39th-minute winner as Italy topped Group A with a 100 per cent record and equalled their national record of 30 games unbeaten.
Switzerland had Welsh hearts racing by establishing a 2-0 interval lead against Turkey in Baku and trimming Wales’ goal difference advantage.
The Swiss eventually won in 3-1, but it was not enough to leapfrog Wales and their qualification hopes now reside on being one of the four best third-placed sides.
Wales instead will head to Amsterdam to play the runners-up of Group B, most likely Denmark, Finland or Russia.
The prize at stake was a last-16 tie at Wembley with Austria or Ukraine the probable opponents.
Italy had established themselves as one of the tournament favourites by overwhelming Turkey and Switzerland, two 3-0 wins making it 10 successive victories by the margin of 31-0.
Manager Roberto Mancini had promised changes with qualification for the knockout stage already secured and made eight in the event.
Wales also altered a winning line-up with boss Robert Page concerned over potential suspensions moving forward.
There were only around 400 Wales supporters present in a Stadio Olimpico crowd of around 14,000 amid tight coronavirus restrictions.
But the travelling fans made themselves heard as the action got under way and Italy set up camp in the Wales half.
Wales did break the stranglehold briefly when Gunter, who has never scored in his long international career, headed Ramsey’s corner just over the crossbar.
But Italy’s superiority was rewarded after Verratti had been bundled over by Joe Allen.
Verratti sent in a low free-kick which Pessina met superbly on the volley to direct the ball past the helpless Ward.
Bernadeschi almost doubled Italy’s lead soon after the restart as his 25-yard free-kick struck a post.
Wales’ task got a lot tougher 10 minutes into the second period when Ampadu brought his boot down on Bernadeschi’s foot and Romanian referee Ovidiu Hategan had no hesitation producing his red card.
Ward saved superbly from Belotti and Bryan Cristante before Bale missed Wales’ best opening 15 minutes from time.
Joe Rodon directed a free-kick onto Bale’s favourite left foot, but the Real Madrid forward snatched at the opportunity and skied the ball over.
It was not to prove costly, though, as Wales held on and maintained their record of reaching the knockout stage of every major tournament they have competed in.