The 37-year-old Ivorian was a free agent following the end of his second spell with the Blues, for whom he scored 164 goals across nine seasons at Stamford Bridge. The deal for Drogba also involves Chicago Fire. The American franchise had filed a discovery claim for the former Galatasaray player but Montreal have now acquired his rights for an undisclosed fee.
Montreal president Joey Saputo told the club's official website: "It's an honour to welcome Didier Drogba to the Impact. "From the first conversation I had with him, I felt that he really wanted to play in Montreal. This is one of the biggest days in club history and his arrival to the club will be beneficial in every aspect."
"He's had an impact on millions of people since the start of his career and he will continue to do so wearing the Impact colours," said Montreal vice-president Nick De Santis. "A player of his magnitude can only help grow our club. He's been a leader, a winner and a champion throughout his career."
The 104-time Côte d'Ivoire international joined Chelsea from Marseille in 2004 and twice scored 20 goals or more in a single league season. He also won the FA Cup four times and the League Cup three times.
Montreal technical director Adam Braz added: "We are very excited and thrilled to add to our roster a player with an exceptional career. He's demonstrated his qualities as a striker at the highest level in Europe. His instinct in front of the goal, his experience and his leadership are all attributes that will help the team on the field."
Frank Lampard and Steven Gerrard are likely to cross paths with Drogba when New York City and Los Angeles Galaxy face Montreal, while former West Ham United and Aston Villa midfielder Nigel Reo-Coker will be his team-mate at the Saputo Stadium. Montreal are currently sixth in the Eastern Conference, one place above Lampard's New York.