Financial details were not disclosed but Marseille president Christophe Bouchet told the French club's web site that Chelsea paid between 30 and 40 million euros (20 to 27 million pounds) for Drogba.
"I am very happy to be here and am looking forward to playing for my new team," Drogba told Chelsea TV on Tuesday after signing a three-year contract.
"I hope it will be a very good and very long adventure with Chelsea and with all the fans of Chelsea," he added.
"I want to do well with my national team and my club, so I will do my best to be better than I was last season with Marseille."
The 26-year-old Drogba is new Chelsea coach Jose Mourinho's fifth signing, joining Dutch winger Arjen Robben, Czech goalkeeper Petr Cech, Portugal fullback Paulo Ferreira and Serbia and Montenegro striker Mateja Kezman at Stamford Bridge.
"We're delighted that Didier has signed and we are certain that he will score lots of goals for us," Chelsea chief executive Peter Kenyon told his club's website.
"He has shown last season for Marseille that he can succeed in Europe's top competitions.
"There's been a tradition at Chelsea of strong, athletic and quick strikers, and I am sure the fans will love him."
Mourinho had said he wanted to complete his transfers before Chelsea's departure for a pre-season
NEW PLAYERS
The Portuguese coach, who replaced Claudio Ranieri as Chelsea coach in June, has now spent around 70 million pounds on new players.
In all, since Russian billionaire Roman Abramovich bought the club last year, Chelsea have spent over 180 million pounds on transfers.
Chelsea's previous record fee was the 17 million pounds they paid Blackburn for Irish winger Damien Duff last year.
Drogba cost cash-strapped Marseille only six million euros when he joined from En Avant Guingamp a year ago.
He scored 18 goals in 35 league games last season and 11 in their European campaign in which they reached the UEFA Cup final.
Drogba, France's footballer of the year last season, is the third most expensive player ever signed by an English club. Manchester United bought England defender Rio Ferdinand for 30 million pounds in 2002 and Argentina midfielder Juan Sebastian Veron for 28 million pounds in 2001.
Chelsea's first game of the season is against Manchester United on August 15 at Stamford Bridge.
Mourinho was further lifted on Tuesday when Iceland striker Eidur Gudjohnsen signed a new four-year contract with Chelsea, who finished runners-up to Arsenal in the premier league last season and reached the Champions League semi-finals.