Coun Burton, who has represented his Bridlington Central and Old Town seat for 12 years, takes the role after Yorkshire's longest serving council leader Steve Parnaby stepped down, after a record-breaking 23 years.
Coun Jonathan Owen will be his deputy.
It comes after the local elections last week which saw the Conservatives strengthen their grip on the council, increasing the number of seats from 47 to 49.
His appointment as council leader still needs to be ratified at the annual general meeting of the council next week, but with the Conservatives holding a huge majority, that should only be a formality.
Coun Burton, who runs a tourism business at Bempton, said: “I want to continue with the great work of the council along with delivering an economic strategy that works for everybody.
"I want to ensure that we continue the great work that Stephen Parnaby has delivered with economic regeneration in our towns and villages and I want to see that move further along."