Lustig, 25, joins Celtic from Norwegian side Rosenburg but has also enjoyed stints with Swedish clubs Umea and GIF Sundsvall.
"He sees himself as a right-back and that is where we see him," Celtic assistant coach Johan Mjallby said in a statement on the club's website.
"Of course, I know him pretty well because he is Swedish," Mjallby continued. "He has played in Norway for a couple of years for a big club. He has a good engine, he is a good footballer and he likes to go forward which is suitable for how we like to play."
Lustig could debut with his new club on Monday when Celtic squares off against Dunfermline.