The Morocco winger, who will join the Blues on July 1 for an initial fee of 40million euros (£33.4m), is to remain at Ajax for the remainder of the season.
It was announced earlier this month that Chelsea had reached a deal with the Dutch club for Ziyech's signature. He is Frank Lampard's first signing as Blues boss.
"I am delighted and proud to have signed for such a huge club as Chelsea," Ziyech told the club's website.
"I am looking forward to next season and hope we can achieve great things together."
Ziyech has laid on more than 20 goals in each of the last three seasons, and has scored 49 times for the Dutch side.
He is viewed as a replacement for Eden Hazard, who left for Real Madrid last summer.
Chelsea director Marina Granovskaia added: "We are delighted Hakim will be joining us, having been a key target of the club's in this last window.
"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."