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Juventus had come from behind to lead Villarreal but were pegged back 2-2 in the final minute of the game.
The La Liga side had started on the front foot and on 18 minutes, Nicolas Pepe slid a delicate ball into the path of Georges Mikautadze, who fired into the bottom corner to put Villarreal in front.
In second half the Bianconeri found the leveller soon after when Federico Gatti thumped an overhead kick into the back of the net.
Conceicao, who had come on at half-time, then raced clear through on goal and slotted past home goalkeeper.
However, Veiga's late intervention for final 2-2 means Juve have only won two of their last 10 matches in the competition (D5 L3).
Villarreal host Manchester City next in the Champions League on 21 October with Juventus travelling to Real Madrid a day later.
