Last week, the PSG captain took part in the Champions League final in a losing effort for the Ligue 1 side who were narrowly defeated 1-0 by Bayern thanks to a solitary goal from Kingsley Coman.
It was already confirmed beforehand that the final would be the Brazilian's last game for the club.
Frank Lampard has been busy in the transfer market this summer, having already recruited Hakim Ziyech, Timo Werner, Ben Chilwell, and Malang Sarr.
The Chelsea manager was keen to add some experience to his defence, though, with the London side linked with a swoop for Silva for around a month prior to him signing on.
The 35-year-old has signed a one-year deal at Stamford Bridge with the option of another.
Lampard received plenty of plaudits for Chelsea's attacking play last season, but came under major criticism for their defensive frailties as they conceded 54 goals in the Premier League - the 12th worse defensive record in the league.
Despite spending £136m this summer so far, Chelsea could flex their financial muscle further, with Kai Havertz or Leverkusen and Jan Oblak of Atletico Madrid both strongly linked with a move to London.