The England worldwide signed a five-year deal with Chelsea and adds experience to a thin midfield after multiple departures this summer.
The 25-year-old Italy global left Rome's Ciampino airport on a private flight for London to undergo a medical early on Thursday.
Torino could accept Chelsea's late bid but they need to find a replacement for Zappacosta before.
Conte will be pleased to have acquired a player on deadline day when at one point it looked as if his frustrations were set to continue.
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Chelsea have completed the signing of full-back Davide Zappacosta from Torino on a four-year deal. "I'm delighted to be a Chelsea player and can't wait to get started", said Drinkwater.
Chelsea have signed midfielder Danny Drinkwater from Leicester on a five-year contract, the Premier League champions have announced.
He is well known to Chelsea boss Conte, who called him up to the senior Italy squad during his time in charge of the Azzurri and will hope he can quickly adapt to the Premier League.