Thiago, 35, joins the Blues on a one-year deal, with the option of extending his stay at Stamford Bridge for another season.
The defender spent almost a decade at PSG before joining Chelsea and helped the French champions reach the Champions League final last week, where they lost to Bayern Munich.
Thiago will bolster a Chelsea defence which conceded the most goals (54) in the Premier League top ten last season and Terry is delighted by his signing.
‘What a signing this is,’ the former Chelsea captain said after the Blues confirmed the deal.
Reacting to his move to west London, Thiago said: ‘I am so happy to be joining Chelsea. I am delighted to be a part of Frank Lampard’s exciting squad.
‘I’m here to challenge for honours and I look forward to playing at Stamford Bridge very soon.’
Chelsea director Marina Granovskaia, meanwhile, said: ‘We are delighted to be able to add a player of Thiago Silva’s proven world-class credentials to our squad.
‘Having performed at the highest level for many years, we have no doubt his experience and quality will complement the many exciting talents we already have here.
‘Thiago will be an excellent fit and we hope he can add some new trophies to his impressive list of honours.’
Thiago becomes Chelsea’s fifth signing of the summer following the arrivals of Timo Werner, Hakim Ziyech, Ben Chilwell and Malang Sarr.