The clash at RAMS Park swung wildly between both sides before the Turkish champions overwhelmed their Italian opponents after the break, turning a narrow deficit into a commanding three-goal advantage ahead of the return leg in Turin.
The hosts made a blistering start and were rewarded inside 15 minutes when Gabriel Sara unleashed a powerful long-range strike that gave Juventus goalkeeper Michele Di Gregorio no chance.
Juventus responded almost immediately. Teun Koopmeiners reacted quickest after a rebound fell kindly to him, levelling the contest just a minute later. The midfielder then completed the turnaround in the 32nd minute, driving home from distance after neat build-up play to send the visitors into the break with a 2-1 lead.
Despite their strong first-half recovery, Juventus' momentum faded quickly once the second period began.
Galatasaray returned with renewed intensity and equalised within four minutes of the restart when Noa Lang finished from close range after persistent pressure in the Juventus penalty area.
The momentum swung firmly in favour of the hosts, who completed their comeback on the hour mark as Davinson Sanchez converted from a well-delivered Sara set-piece to put Galatasaray ahead.
Juventus' hopes of responding suffered a major blow when substitute Juan Cabal was dismissed after receiving a second yellow card just over 20 minutes after entering the pitch.
With the visitors reduced to 10 men, Galatasaray pressed relentlessly to extend their advantage. Lang grabbed his second goal of the night, punishing defensive hesitation inside the Juventus box after a precise assist from Victor Osimhen.
Osimhen was again influential late on, setting up substitute Sacha Boey, who drilled home from a tight angle to complete the scoring and cap a dominant attacking display.
Guided by coach Okan Buruk, Galatasaray's relentless second-half performance leaves them in a commanding position as they chase a place in the last 16 for the first time in more than a decade. Meanwhile, Juventus boss Luciano Spalletti faces a huge task to revive his side's campaign when the teams meet again in Turin.
Galatasaray will carry a significant three-goal cushion into the decisive second leg, leaving Juventus needing a dramatic turnaround to keep their European ambitions alive.