Farbrace, a former assistant manager of the Sri Lanka side who has been coaching Yorkshire's second XI, has agreed a two-year contract, Sri Lanka Cricket (SLC) said in a statement.
"The final decision was made after much deliberation as a selection of Sri Lankan candidates showed commendable credentials as applicants to this post," the SLC said.
"SLC wishes to thank Yorkshire Cricket Club for being generous enough to release Farbrace for duty in Sri Lanka," it added.
Ford, who took over when Australian Geoff Marsh was sacked in January 2012, had decided to step down after the test series against Pakistan in the United Arab Emirates starting later this month.
Sri Lanka's assistant coach under Trevor Bayliss from 2007 to 2009, the 46-year-old Farbrace will be in charge of the team's World Cup campaigns both in the Twenty20 (2014) and 50-over (2015) formats.
"Paul is an outstanding coach and deserves to take on this role with Sri Lanka Cricket," said Yorkshire's director of cricket Martyn Moxon.
"He has implemented a successful coaching model, which has underpinned our success over the past two years.
"He is a good friend and will always be welcome to visit us at Headingley at any time."