The Chilean struck two penalties either side of Olof Mellberg's equaliser, before heading his third to complete his first ever treble and ensure his side claimed maximum points for the first time in the competition this season.
The result means Juve sit second in Group B, two points clear of Galatasaray and needing a point from their trip to Turkey in a fortnight's time to progress to the knockout stages.
After the game, Vidal was understandably delighted with his performance.
"It's the first time that I've scored three goals in a match in my career so I'm happy, especially because we won," he said.
"After they equalised we tried to score as soon as possible because we wanted to control the game, and we managed it even though it wasn't easy because of the way they defended."
Vidal went on to reveal that he would be taking the match ball home to his son, the hat-trick coming as an extra bonus for him and his family with another baby on the way.
"I'm taking the ball back home," he said.
"It's a gift for my son, to show him when I played this game. We are expecting little girl now, so I'm very happy."
There were doubts before the game that Vidal would even play, the midfielder being one caution away from suspension, but he said afterwards that Antonio Conte felt the Copenhagen match too important for him to miss.
"There was some fear," he said.
"We knew this was an important game for us, but the coach told us the important thing was to win, so I could play."
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