The 37-year-old – whose first language is French – will be playing in French-speaking Montreal
He left Stamford Bridge this summer after playing 381 games and winning four Premier League titles across two spells with the London club.
Drogba is Cote d’Ivoire’s all-time top scorer, with 65 goals in 105 international appearances.
Montreal president Joey Saputo said the signing made it “one of the biggest days in club history”.
He added: “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.”
Drogba will be a designated player at the Canadian side, which means his wages will not be restricted by the MLS salary cap.
Impact, MLS members since 2012, finished bottom of the table last season.
They secured Drogba’s services after completing a trade with Chicago Fire for the right to negotiate a free transfer with the Ivorian star, who agreed an 18-month contract.
The Canadian side sent Fire allocation funds – consisting both of Targeted Allocation Money (TAM) and general allocation money – in order for the club to drop their own claim on the 2012 Champions League winner.
Drogba has been a target of MLS for several years, after allegedly turning down a $7.75 million-a-year deal to sign with the league back in 2012 before he accepted a lucrative offer to move to China that ultimately collapsed.
He follows former Chelsea team-mate Frank Lampard in moving to the North American league, with the 37-year-old having joined New York City.