The Serie A leaders held Luis Enrique's side to a 0-0 draw in Wednesday's second leg to secure a 3-0 aggregate victory and a place in the semi-final draw.
Neymar was in tears after the final whistle and was comforted by his compatriot and former Barca team-mate, as well as Juve forward Juan Cuadrado.
Dani Alves says he told Neymar that the defeat was simply part of football and that he should embrace it as a valuable experience.
"I told him that it's the way things work. It was fate that saw us face each other," he said.
"I didn't want to play them for the sentimental value that surrounds this game, for all that we went through. We all live it together and it's hard once you're separated.
"Sentiments have to be put to one side. You have to do your job and defend your colours with honour, but the taste is still there. It's how it goes.
"Today you win, tomorrow you lose, but it's not a great defeat. Neymar has to look at it as a learning curve. You have to try to lift yourself after these situations."
The 33-year-old later posted a picture of his embrace with Neymar, captioned simply: "I love you brother".
Juve will face one of Real Madrid, Atletico Madrid or Monaco in the last four.