Vicario, who joined on loan from Tottenham this summer, kept a clean sheet in a home victory built on Nico González's deflected strike and a late Teun Koopmeiners finish.
Reflecting on the occasion afterwards, the Italy international struggled to find the words.
"Putting it into feelings is complicated for me, because they're emotions and vibrations that are hard to describe. It's what I've dreamed of my whole life - I was made for this dream. Today I made a dream come true," he said.
"For tonight I'm happy, but the journey has only just begun. There are many more nights like this still to come, and we need to work with enthusiasm to have more of them. Juventus played a good match - we were patient and got a good win."
Asked to reflect on his first weeks in Turin, Vicario pointed to a smooth transition despite an unconventional pre-season.
"I already knew the manager from the national team, and I knew his methods. I'm coming from a different league, the transfer market took longer than I expected, but I've settled in well, even without a normal pre-season. I've got enthusiasm and energy for it," he said.
On sharing goalkeeping duties with new arrival Kamil Grabara, signed on loan from Wolfsburg, Vicario was warm in his praise: "Grabara is doing great, he's a strong, capable goalkeeper - we help and push each other. We represent Juventus, and the level has to be high."
On what head coach Luciano Spalletti had specifically asked of him since his arrival, Vicario kept it simple: "To do the simple things, and what I know how to do. He asked me to be myself."
Vicario's move to Juventus followed two seasons in England with Tottenham, where he established himself as one of the Premier League's most reliable shot-stoppers before returning to Italian football this summer.
His arrival prompted a significant reshuffle in Juventus's goalkeeping department, with Michele Di Gregorio leaving on loan for Bournemouth and Mattia Perin heading to Palermo, while Carlo Pinsoglio remains on hand as third choice.
Vicario briefly went off during last weekend's opening win at Frosinone after suffering cramp, but manager Spalletti confirmed afterwards there was no muscle injury, and the goalkeeper was back in his usual place between the posts against Parma without any fitness concerns.