Matthews spent six years at the helm at BAA – which was renamed as Heathrow Airports Holdings in 2012.
He will take over the post of chair of the agency from Alan Cook in September and be paid £130,000 for a minimum of 78 days work a year.
Matthews will oversee the transformation of the agency into a government-owned company.
Secretary of State for Transport Patrick McLoughlin said: “Colin Matthews’ appointment demonstrates the commitment we have to revolutionising our roads.
“His strong business record will give the new strategic highways company the leadership and focus on customer service it needs, which means taxpayers and road users will get a network fit for the future economic demands of this country.”