Juventus, however, showed in the opening stages that they are made of sterner stuff than PSG, who gave up a 4-0 first-leg advantage.
Massimiliano Allegri's side pressed Barca in their own half and limited the supply to the front three.
Messi almost managed to pick out Jordi Alba with a hooked pass and later played a one-two with the Spaniard before dragging narrowly wide of the far post.
Ivan Rakitic and Neymar also missed the target with hopeful efforts but Juventus' aggressive game plan limited how much attacking Barca could do without leaving themselves exposed.
Gonzalo Higuain cracked a shot over the bar before the break while the probing Juan Cuadrado almost picked out the Argentine striker in the middle, but was thwarted by a brilliant block from Gerard Pique.
Barca increased their grip on possession after the break and spent most of the remainder of the game in their opponents' half trying to carve out chances, but Juve were always ready for them.
Messi whipped a free kick onto the roof of the net and slammed another shot wide before spurning his best chance, a volley which fizzed over the bar.
It was Barcelona's second successive elimination at the quarter-final stage which will provoke much soul-searching at the five-times European champions.