Ajax announced last week that the 26-year-old would be moving to Frank Lampard's side in a deal potentially worth €44million.
The forward, who will spend the rest of 2019-20 with the Eredivisie champions, told Chelsea's official website: "I am delighted and proud to have signed for such a huge club as Chelsea. I am looking forward to next season and hope we can achieve great things together."
Chelsea director Marina Granovskaia made clear the club's delight at signing a player who has taken the Champions League by storm in recent seasons.
"We are delighted Hakim will be joining us in the summer, having been a key target of the club's in this last window," she said.
"He has consistently been one of the most dangerous attacking players in Europe in recent years, which we saw first-hand in our two games against Ajax in the Champions League.
"We wish Hakim and Ajax all the best for the rest of the season and look forward to welcoming him to Stamford Bridge in the summer."
Ziyech joined Ajax from Twente in August 2016 and was a star performer as Erik ten Hag's men reached the semi-finals of the Champions League and secured a domestic double last term.
The Morocco international is the first major signing confirmed by Chelsea since Lampard took over as head coach from Maurizio Sarri prior to the start of the season.
Deals to bring in Christian Pulisic and loanee Mateo Kovacic permanently had been agreed prior to Lampard taking charge, and the Blues were then forced to endure a one-window transfer ban after they were found guilty of breaching FIFA rules regarding the signing of minors from overseas.
Their initial two-window suspension was reduced by the Court of Arbitration for Sport (CAS) in December, however, allowing Chelsea to sign players in January, but they failed to secure any new arrivals.
Speaking at the start of this month, Lampard said his side had become "underdogs" in the battle to finish fourth in the Premier League because their rivals had strengthened their own squads.