Dybala netted twice before the midway point in the first half and Giorgio Chiellini added the third goal early in the second half in Turin to leave Juve in control of the tie ahead of the next Wednesday's return at the Camp Nou.
Juve-Barca, Bayern-Real in Champions League quarter-finals draw
Having made history when they overturned a 4-0 first-leg deficit against Paris Saint-Germain in the last 16 by winning the return 6-1, Barcelona will need another stunning comeback if they are to reach the last four.
Juventus last met the Spanish giants when they lost 3-1 in the 2015 final, but here they ran them ragged and nullified the visitors' front three of Messi, Neymar and Luis Suarez.
Emotions to run high as Alves faces former club Barcelona
But Barca's feat against PSG means Juve coach Massimiliano Allegri is not getting carried away.
"I want to congratulate the lads because, as a team, they did great," said Allegri. "It isn't easy overcoming a team like Barcelona, but we also dug deep to keep a clean sheet. That was fundamental for us.
"But we have to remain humble, keep our heads down and keep working. PSG scored four, and look what happened."
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