The defender signed a four-year deal with the west London club, while the England global agreed to a five-year deal with the English champions.
Chelsea then confirmed the 25-year-old Zappacosta signed a four-year contract, becoming Chelsea's fifth signing of a hard summer transfer window.More news: 'Hats off to Tamim and Shakib'
Both deals were concluded in the final hours of the transfer window, strengthening a Chelsea squad that had looked thin early on this season.
Chelsea have signed Davide Zappacosta from Torino, it was confirmed on Thursday.
Antonio Conte has been desperately trying to bring in a full-back all summer but the Blues boss has been frustrated with his pursuit of several key targets in recent weeks.More news: Liverpool Offered Close to $107million to Sign Lemar
Drinkwater told his new club's website: "I'm delighted to be a Chelsea player and can't wait to get started".
Drinkwater played a big role in Leicester's incredible title-winning campaign of 2015/16.
Zappacosta has signed a four-year deal with the Premier League champions - in a move that sees him link up once again with Antonio Conte, who selected him for Italy's Euro 2016 squad.More news: World badminton PV Sindhu wins the Silver Medal
The Premier League champions have turned their attentions to Zappacosta after losing out to Liverpool in the race for Arsenal's Alex Oxlade-Chamberlain. I am very excited to be here and start working with the team.