Alberto De Rossi's side, currently atop the league standings as the Under-19s reach the final third of the campaign, were forced to wait until the death to clinch victory in Turin - as Davide Mastrantonio saved two penalties to ultimately send his side through.
"We really wanted this, it's been too long since we progressed a good way in the Coppa Italia," Mastrantonio said.
"It's all instinct in the moment. You react to how the opponent looks, which foot he's going to kick with ... and that's it.
"It was a great game, with both teams attacking and counter-attacking. Perhaps we created more chances overall but it was a good game where both sides played well."
The Giallorossi went behind early on, only for Antonio Satriano to equalise after latching onto Joel Voelkerling Persson's low cross.
In the second half De Rossi's side went ahead - Riccardo Pagano with the finish - only to be pegged back themselves this time.
That set the stage for extra-time and then penalties, as Roma converted four of their five attempts to win the shootout 4-3.
"Another great performance where the lads really showed the character they have," De Rossi said afterwards.
"They've done that a lot this season - maybe not in every game, but certainly in the majority of them."
He added: "We had to overcome a lot of obstacles, before the game and during it too. Juventus are a good side but I always thought we showed a little bit more and we deserved to progress."
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