The Chilean converted two penalties either side of Olof Mellberg's equaliser before nodding in a third to take his tally to five goals in the competition this season.
The result, which eliminates Copenhagen, leaves Juventus second in Group B, two points clear of Galatasaray, whom they visit in their final game with just a point needed to progress to the knockout stages.
Vidal's first penalty came shortly before the half-hour mark when Lars Jacobsen handled, but the hosts were momentarily stunned in the 56th minute when Mellberg levelled the scores against his former club.
Midfielder Vidal claimed his second from the spot five minutes later following a foul by Mellberg before completing his hat-trick with a looping header two minutes later.
The hosts made one change from the weekend as Leonardo Bonucci replaced Claudio Marchisio, while Claudemir and Christian Bolanos came into the Copenhagen side in place of Cesar Santin and Igor Vetokele respectively.
Paul Pogba wasted the first chance of the game in the seventh minute, firing a weak shot at Johan Wiland from the edge of the box after beating the offside trap.
In the 29th minute, Juventus were gifted the chance to break the deadlock when Jacobsen inexplicably handled a cross while under pressure from Pogba and the referee had no hesitation in pointing to the spot.
Vidal stepped up and slammed the penalty low to Wiland's right, his third successful spot-kick in this year's competition.
Juve were dominant as the half wore on, Pogba rippling the side-netting from the edge of the area four minutes before the break.
Copenhagen equalised 11 minutes into the second half when Mellberg hooked the ball home from 10 yards after the Juve defence failed to clear a long throw from the left.
Parity lasted just five minutes, however, Mellberg's evening turning sour as he hauled down Fernando Llorente inside the area, Vidal once again sending the ball low into the bottom left-hand corner.
Vidal put some daylight between the two sides two minutes later, hanging in the air at the far post to meet Pogba's left-wing cross and loop a header into the far corner.
The remainder of the game proved uneventful with Juventus happy to sit back and soak up pressure to claim their first win of the campaign at a crucial time.
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