The Dubliners, who only had to pay for the services of two members of their starting line-up, were beaten by a goal in each half from Amauri, a player who cost twice European Cup winners Juve over 20 million euros.
The Brazilian-born striker, a big money capture from Palermo two years ago, ended Rovers unlikely hopes of an upset by poking home after a neat one-two on three minutes and heading in a second 70 minutes later.
Northern Irish winger Tommy Stewart, a Juventus fan who scored the winner against Israel's Bnei Yehuda in the last round, continued to trouble the visitors down the left but his side just could not find a breakthrough.
Amauri, who received Italian citizenship in April but was not picked for the dreadful World Cup campaign, nodded in his second to ensure there would be fewer nerves in next week's return leg.
There was enough time for another couple of half chances for the hosts and also for Alessandro Del Piero to enter the fray to a standing ovation from some and sarcastic chants of "you'll never play for Rovers" from others.
"Overall I am happy. The team performance was very good," Del Neri told a news conference through an interpreter, a local Italian restaurant owner called in to translate.
"We're not going to go," Shamrock Rovers boss Michael O'Neill joked when asked how he would approach the second leg.
"I'm very proud of the players. They can all be very proud of themselves having never played against players of that calibre before... You certainly don't see players like that in the League of Ireland."
