The 21-year-old, who is the younger brother of Eden Hazard, made the switch to Stamford Bridge from Ligue 1 side Lens in 2012.
Chelsea immediately opted to loan the younger Hazard to Zulte Waregem in his native Belgium, where he has spent the past two seasons.
Gladbach claimed to have reached an agreement to loan the player on Thursday, and the temporary deal has now been confirmed.
Max Eberl, Gladbach's sporting director, told the club's official website: "Thorgan is a promising talent.
"He can be used in various offensive positions and we are pleased that he has decided to take the next step in his career at Borussia Monchengladbach."
Hazard's two seasons with Waregem proved fruitful, and last season he won the golden shoe for the best player in Belgium's Pro League.
He was rewarded with a first international cap last May, and was on Belgium's standby list for the FIFA World Cup.
