Sarri will arrive having left Napoli in May while Zola's most recent role came as manager of Birmingham City between December 2016 and April 2017.
Conte took training when some of the Chelsea squad returned for pre-season this week but has always been expected to be replaced this summer.
Not getting into the Champions League was a big minus to Chelsea.
"It is an exciting new period in my career", said Sarri. I look forward to starting work and meeting the players on Monday ahead of travelling to Australia, where I will be able to get to know the squad and begin our match action.
Sarri guided Napoli to a second place finish in Serie A last season with an astonishing 91 points, narrowly missing out on the Scudetto to Juventus.
Sarri will become the 12th manager to take over the reins at Chelsea since Roman Abramovich bought the London club in 2003.More news: Elon Musk Calls Diver Involved in Thai Rescue Efforts a ‘Pedo Guy’
"We are talking about a lad for whom Manchester City represents the opportunity of a lifetime".
Chelsea has not made any major summer signings in the summer transfer window so far because the board is waiting to confirm the new boss at the club.
"He has demonstrated his quality working under Maurizio Sarri and we are convinced his signing will help bring further success to Stamford Bridge".
Should Chelsea have to sell Eden Hazard to Real Madrid this summer, they will be back in the market for a winger.
The Italian ex-professional footballer had managed Chelsea FC for two years before his contract was terminated.
Winning trophies didn't save Conte being shown the Stamford Bridge exit door as he was dismissed after guiding Chelsea to the FA Cup in May.
He will take over a Chelsea side who finished fifth in the Premier League last term, but won the FA Cup.