Zappacosta has four full Italy caps and has been at Torino for two seasons, having previously played for Atalanta.
He has been frustrated during much of the summer transfer window, losing out on Alex Oxlade-Chamberlain to Liverpool and long-term target Fernando Llorente to London rivals Tottenham.
Conte knows all about Zappacosta, having given him his first call-up to the Italy squad previous year during his time as national team manager. "I'm delighted to be a Chelsea player and can't wait to get started", said Drinkwater. "I hope I can show my quality to the coach, my teammates and the supporters", Zappacosta told the club's official website.
"We are very pleased to be able to bring him to the club and believe he will become an important member of the squad".
And, they could have been joined by a few more players too, with Everton playmaker Ross Barkley pulling an outrageous U-turn during his Stamford Bridge medical.
Danny Drinkwater and N'Golo Kante's midfield partnership was crucial in Leicester City's title triumph in 2016.
News broke earlier on Thursday that the Italian worldwide was flying-in to London to undergo his medical and discuss personal terms after Chelsea had agreed a deal with Torino for his signature.