PSG sporting director Leonardo joined Alex in addressing the media in a press conference broadcast live by PSG TV.
The 29-year-old joined Chelsea in 2004 but has struggled for first-team football under Andre Villas-Boas this season and has now been reunited with former Blues boss Carlo Ancelotti.
French sports newspaper L'Equipe reported that the transfer fee was 5million euros (£4.18million).
PSG later confirmed on their website that Alex had signed a two-and-a-half-year deal.
Chelsea also confirmed the move on their website, saying: "Chelsea Football Club can confirm we have agreed terms for the permanent transfer of Alex to Paris St Germain.
"Everybody at Chelsea Football Club would like to express their thanks to Alex for his service and we wish him well for the future."
