From April, workers aged over 25 will receive a minimum of £7.20 per hour.
The Regulatory Policy Committee, which advises government, estimates the change will cost companies £804.4m in extra wages and staff costs.
A further £234.3m of "spillover" costs from keeping pay differentials will take it over £1bn, the body said.
The amount is slightly higher when including the public sector, where more workers are already paid above the National Living Wage.
According to the Institute of Directors (IoD), the figures show George Osborne must now "come good" on his promise to cut taxes.
"IoD members supported the introduction of the Chancellor's living wage as part of a deal he made with business - lower taxes for higher wages," said IoD director Seamus Nevin.
He said companies would pay £12bn over the next five years for the government's new apprenticeship levy, as well as further costs from pensions auto-enrolment and extra reporting requirements.
Mr Nevin added: "It is imperative that the government now comes good on its promise of less red tape, fewer regulatory hurdles, and a lower rate of corporation tax to help employers absorb these additional costs and raise pay."
The National Living Wage is separate to the voluntary living wage, which is set at £8.25 per hour and £9.40 an hour in London.
More than 2,000 businesses, with almost 70,000 workers, are signed up to the latter scheme.
UK businesses face more than £1bn in costs from the introduction of the National Living Wage next year, according to government advisors.