The Turkish Super Lig leaders secured their place in the play-offs after ending in 20th place during the initial league phase of the competition as the fifth-placed Serie A giants ended in 13th.
Gabriel Sara opened the scoring for Okan Buruk’s charges after 15 minutes, before a quick-fire Teun Koopmeiners equaliser a minute later.
Koopmeiners put Juve in the lead ahead with 32 minutes played but Noa Lang levelled with his goal five minutes after the half-time break.
Goals from Davinson Sanchez (60′), Lang (74′) and Sacha Boey then sealed the win after Juve were reduced to 10 men, with Juan Cabal dismissed with a second yellow, on 67 minutes.
The result left Galatasaray in a commanding position ahead of their second leg on Wednesday 25 February, while leaving Luciano Spalletti’s team with a mountain to climb.
Speaking after the game, Spalletti admitted he was disappointed with the result, and with the red card to Cabal, having only brought him on at half-time as a replacement Andrea Cambiaso, who was shown a yellow card in the opening period.
The coach said on beinsports: “We ended the first half badly, tried to get back into shape and changed a few things, but we really dropped off in terms of character.
“It was not one step back, but three steps back this evening. Clearly, (Andrea) Cambiaso risked a second yellow card at the end of the first half.
“In these games, when you have someone on a card, it is wise to do something with the substitutions.
“However, we received two more yellow cards in the first 15 or so minutes, then we did further damage to ourselves by not recognising the danger in those situations.”
Spalletti added: “I want us to create something going forward, not just breaking up the play. Not every situation is the same, sometimes you kick it away and release some pressure. I like to play the ball in general, but there are different situations.
“When down to 10 men, that’s a difficult situation. We made the substitution to take pressure off one player and his replacement received two yellow cards in 20 minutes. That makes everything tougher.
“Quite apart from the cards, the whole team took a few steps back in various ways today.”
Meanwhile, a delighted Buruk expressed his happines with the result, adding: “First of all, of course, we are incredibly happy. It is a night where we need to keep our feet on the ground. We achieved an important victory.
“There is significant effort, not just from us, but also from our fans. We had a very beautiful night, and we created it.
“For the players, it is a very important night for me and for Galatasaray. We need to play the same way in the second match to crown this.
“After playing the second match, we need to celebrate. Right now, there is confidence and happiness. We will be even happier after passing the round.”