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.
His early strike briefly silenced most of the 5,800 supporters crammed into the tiny west Dublin stadium but the hosts made former Sampdoria coach Del Neri sweat for his victory by enjoying long periods of possession.
While the only other highlight of first half was a 20-man scuffle just before half-time, Rovers should have equalised six minutes after the break but captain Dan Murray headed wide from close range.
The home side, midway through their pursuit of a first domestic league title since 1994 and looking sharper as a result, survived a scare a few minutes later when Amauri hit the post with an effort almost identical to his opener.
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.
Juve, who finished seventh in Serie A last term after one of their worst top-flight seasons, will host the August 5 second leg in Modena because their Turin home is being used for a concert by some far more famous Irishmen, rockers U2.
"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."
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