Sergio Busquets found Lionel Messi with a clever ball into the area and the Argentine carved out space on the left to deliver a lofted cross to Luis Suarez at the back post.
The Uruguayan's header deflected off the shoulder of a scrambling Shaw and beat David De Gea to give the visitors the lead and although the linesman initially flagged for offside, the goal was given after a VAR video review.
United's best chance of the first half fell to Diogo Dalot but the Portuguese made a mess of a fine Marcus Rashford cross, heading harmlessly across the goal from close-range.
Ole Gunnar Solskjaer's team had struggled to get a foothold in the game during the opening 45 minutes but they were much improved after the break, applying plenty of pressure without creating any clear chances.
Barcelona could have added a second had Suarez not fired into the side-netting from a promising position after a clever through ball from Nelson Semedo.
Ronaldo gives Juventus edge against Ajax
Cristiano Ronaldo scores Juventus’s first goal - Champions League Quarter Final First Leg - Ajax Amsterdam v Juventus - Johan Cruijff Arena, Amsterdam, Netherlands - April 10, 2019. (Photo: Reuters)
Meanwhile, Cristiano Ronaldo produced yet another quality goal to add to his collection and help Juventus to a 1-1 draw at Ajax Amsterdam in the first leg of their Champions League quarter-final tie the same day.
Juventus took the lead on the stroke of halftime as Portuguese forward Ronaldo began an attacking move inside his own half before sprinting down field and diving to head in a clever chip from Joao Cancelo.
Ajax equalised 30 seconds into the second half after Cancelo lost control of the ball to allow David Neres to run down the left wing, cut inside and curl the ball past Juve goalkeeper Wojciech Szczesny.
The result leaves the Serie A side in a strong position to advance to the semi-finals in next week's return leg in Turin. However, they will be mindful of Ajax's 4-1 away win over Real Madrid in the last round that eliminated the holders 5-3 on aggregate.
Hakim Ziyech had several efforts at goal in a lively start for the youthful Dutch team, who have come through the pre-group phase qualifying round to reach the last eight.
Donny van de Beek scraped the outside of the upright with a 25th minute effort from close range after opening up space for himself in the penalty area in Ajax's best chance of the first half.
But Juventus, with the 34-year-old Ronaldo central to their attacks, worked their way into the contest and came close themselves when the forward set up Federico Bernardeschi to turn and shoot in the 37th minute. But he was just off target.
Ronaldo, who scored a hat-trick in the previous round against Atletico Madrid, then scored a 125th Champions League goal with the last move of the first half.
