The MLS side’s president Joey Saputo stated that the transfer was “one of the biggest days in club history”.
“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.
“His arrival will be beneficial in every aspect.”
The 37-year-old will be a designated player at the MLS side, meaning his wages will not be restricted by the salary cap.
Drogba spent two seasons at Galatasaray and scored 20 goals in 53 games before joining Chelsea in the summer.
The Ivory Coast international won the Süper Lig, Turkish Cup and reached the quarter-final of the Champions League while at the Lions.
Drogba provided two assists and seven goal in 40 appearances for Chelsea last season.
The former Galatasaray striker is currently a free agent after his contract with Chelsea expired in the summer. Drogba has won the Premier League on four occasions.