The Hoops took everything Bnei Yehuda could throw at them on an intensely hot night, but with Alan Mannus in outstanding form in goals, they took their chance and set up a mouth-watering tie with the Italian giants.
Juve will visit Tallaght next Thursday, with the return leg set for Modena and the celebrating Rovers fans at the Bloomfield Stadium could scarcely believe their eyes.
Michael O'Neill made one change to his starting line-up from the first leg as Robert Bayly started ahead of Paddy Kavanagh.
With Gary Twigg operating as a lone frontman, they started the better team and but for the crossbar they could have been ahead within the opening four minutes.
The Israelis looked uncomfortable under the high ball during both legs and Craig Sives beat goalkeeper Dele Aiyenugbha to Enda Stevens' ball, only for Kfid Ederi to clear for a corner.
And from the set-piece, James Chambers found Dan Murray and the skippers' downward header came off the ground and struck the crossbar.
Having made the early running, the Hoops were forced to chase the ball for long stretches as Yehuda kept possession.
That made life difficult in the heat, and cracks began to appear in the Rovers defence. Mannus had to snuff out a chance at the feet of right back Yitzhak Azuz, before he did well to deny Oz Ezra Raly who was played through by the creative Liroy Zhairi.
The Northern Ireland stopper could only watch as Linig Sinisa Hvratsk's shot took a tricky deflection off Egiya Yavruyan, but spun just wide of the post while Yavruyan had a go himself only to see the ball go just wide.
A break for an injury to Stephen Rice allowed the Rovers players take on water and they looked revived and finished the half strongly, Twigg going close twice.
Scoreless at the break, it was the Irish side who needed a goal but the Israelis who looked like getting it.
But Mannus came to the fore. First he made a point blank save from Zhairi's header after Ezra Raly got to the by-line.
He then reacted brilliantly to tip Sinisa Hvratsk's rising drive over the bar as the pressure looked like telling and he dived at the feet of Ezra Raly.
But it was Rovers who made the breakthrough. Rice's long throw caused havoc in the Yehuda defence. Aidan Price nodded the ball into the danger zone and with Ivan Garrido hesitant, Stewart took full advantage to nip in and score.
The Israelis threw everything into getting an equaliser but the closest they came was when Murray cleared Yavruyan's header off the line. They had chances to extend their lead, but one was enough.
Bnei Yehuda: Aiyenugbha; Azuz (Baldut 79), Garrido, Viktor Hadad, Ederi; Abu Zaid (Levi 90), Zhairi, Sinisa Hrvatsko (Menashe 74), Ezra Raly; Hafike, Yavruyan.
Shamrock Rovers: Mannus; Sives, Murray, Price, Stevens; Chambers (Dennehy 64), Rice, Turner (Bradley 84), Bayly (Kavanagh 58), Stewart; Twigg.
Referee: Antony Gautier (France)