The 21-year-old Belgian – brother of Chelsea star Eden Hazard – is currently on loan at the Bundesliga side and has signed a permanent deal that could keep him at the club until 2020.
The size of the fee isn’t known, but Chelsea are believed to have included a repurchase option of €8million (£6million) in the transfer agreement.
“Thorgan has indicated in the past few months what a great talent he is,” said Borussia sporting director Max Eberl. “We are delighted we have managed to sign him, and that he has decided to play for the next few years for Borussia.”
Hazard said: I signed here for four years plus a one-year option because I think Borussia is the right club for me to keep growing.
“It is too soon for me to play for Chelsea. Borussia is a club that suits me perfectly. Monchengladbach is not far from Belgium and the club has great fans that always support us. They are a good team, have a good coach and it is a good place for me to develop.”
Hazard has made 24 competitive appearances for Borussia in 2014-15. He scored four goals and chipped in with seven assists before Christmas.
The Belgium under-21 international spent two seasons on loan at Zulte Waregem after signing for Chelsea in 2012.
The news comes just two weeks after Hazard’s brother Eden Hazard signed a new deal at Chelsea worth £200,000 a week.