The Hoops have been keen on the Sweden international for the past few weeks and have now clinched their man who will arrive at Parkhead on January 1.
24-year old Lustig, who has also attracted recent interest from Fulham and Espanyol, will officially sign a three-year deal when he lands in Glasgow and he is delighted to be joining the Scottish giants.
"This is a really exciting time for me and I am delighted to be committing my future to Celtic, " Lustig told the club's official website.
"I know a lot about Celtic, about the size of the club, the world-famous supporters and about the quality squad of players which is already there.
"I'm really looking forward to joining this squad, to the challenges ahead and hopefully playing my part in bringing some success to the club."