Hazard, the younger brother of Stamford Bridge star Eden, joined the Blues in 2012 from Lens and spent the last two seasons back at home in Belgium with Zulte-Waregem.
He told his club's official website borussia.de: "First of all I am happy that I can now play for Borussia. I do not know too much about the club, but I hope that I will soon learn more about it.
"I know that (Germany international) Marco Reus has played here and of course I also know that my fellow countryman Filip Daems (plays there). Everything else I will certainly learn in the coming time.
"I now must also get to know my new team, so it's perhaps a little too early to talk about it.
"I'll try to do my best and to attend as many games as possible. That's the most important thing for me."
He impressed enough to earn his first senior cap for Belgium and was recognised with the Golden Shoe for the best player in Belgium in January 2014 before capping his season by winning the Professional Footballer of the Year award.
Hazard scored 14 goals in the Pro League last season and added a further 17 assists - he was also part of the team's Europa League squad, although they failed to get out of the group stage despite taking four points off Wigan.
The Bundesliga will provide a step up in class for Hazard, and Monchengladbach have a lot to live up to after finishing last season in sixth place, enough to earn them a place in the Europa League play-off round.
Monchengladbach sporting director Max Eberl said: "Thorgan is a promising talent. He can play across several positions in attack and we're happy that he's decided to take the next step in his career at Borussia."