Former Juventus defender Olof Mellberg swiftly went from hero to villain for Copenhagen when he equalized in the 56th before giving away a penalty four minutes later when he brought down Fernando Llorente.
Vidal again converted and headed home two minutes later to complete his hat trick.
"I am very happy, it's my first hat trick in my career," Vidal said. "The match ball is a present for my son. I'm always a happy person. Pulling on the Juventus shirt and going out onto the pitch is very important to me.
"I've played in every position apart from the goalkeeper's. But if the coach wants, I'll even play in goal," he joked. "We're very close to signing a new contract. I'm waiting for the club, but I'd even sign one tomorrow."
Juventus moved second in Group B, two points ahead of Galatasaray, which lost 4-1 to 10-man Real Madrid.
The home side had its first opportunity in the sixth minute when Paul Pogba carved out some space for himself at the edge of the area, but his effort was too weak and Copenhagen goalkeeper Johan Wiland was able to gather it comfortably.
Carlos Tevez volleyed wide shortly after. The Juventus striker was furious with the referee for awarding a goal kick as his shot had been deflected past the left upright.
Juventus was pressing but failing to create clear-cut opportunities against a Copenhagen side rooted mainly inside its own half.
Knowing they needed to win, Conte's men began to look nervy as the minutes ticked by before a moment of madness from Jacobsen, which saw Andrea Pirlo's ball for Pogba land on the Copenhagen defender's outstretched arms.
Wiland guessed the right way, but could do nothing to prevent Vidal's powerful penalty from going into the bottom right corner.
Pogba went close to doubling Juventus' lead shortly before halftime when he curled an effort into the side netting.
Copenhagen almost leveled immediately after the break as the ball ricocheted around the six-yard box but Ragnar Sigurdsson failed to get the final touch.
Juventus was looking dangerous as first Tevez blasted over before Kwadwo Asamoah had a shot blocked.
However, it was Copenhagen which scored, with its first shot on target, as the Juventus defense failed to deal with a throw-in and Melberg headed home. The defender, who spent a season at Juventus, didn't celebrate.
Melberg gave away a penalty on the hour mark, when he wrestled Llorente to the ground. Vidal again fired into the bottom left corner, this time sending Wiland the wrong way, to restore Juventus' lead.
The Chile midfielder became the third Juventus player to score a hat trick in the Champions League after Filippo Inzaghi and Alessandro Del Piero when he sent a looping header into the far top corner from a Pogba cross.
Vidal was given a standing ovation when he was substituted seven minutes from full time.
