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 next Wednesday's return at the Camp Nou.
Having made history when they overturned a 4-0 first-leg deficit against Paris SaintGermain in the last 16 by winning the return 6-1, Barcelona will need another stunning comeback if they are to reach the last four.
"We basically gifted two goals to Juventus in the opening half, and as the coach, for me, that's unexplainable," said irate Barcelona coach Luis Enrique.
"It's like a nightmare. We've had very little luck of late, and now I can only hope that from tomorrow we get back on our feet."
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.
