The Australia international finalised his move following his recent stint at Celtic's mid-season training camp in Spain.
Fulham, Queens Park Rangers and Reading had also shown interest in the 20-year-old who has been capped four times by his country.
Rogic is Celtic's second signing this month following the acquisition of Israel international defender Rami Gershon on loan from Standard Liege until the end of the season.
In a statement Celtic said: "We are delighted to announce the signing of Australian Tom Rogic on a four-and-a-half year deal, subject to a work permit
"The club can confirm an application will be made to the appropriate authorities in the next couple of days."