The proposals - if they pass into law - would make the UK the first country in Europe to ban cigarette sales to younger people.
Denmark is considering a similar move.
They could phase out smoking among young people almost completely as soon as 2040, a government briefing paper with details of the proposals said.
"A 14-year-old today will never legally be sold a cigarette," Sunak told the Conservative Party conference of his proposed new rule.
Under his anti-smoking plan, Sunak said the smoking age would be raised by one year every year, meaning a younger generation could grow up "smoke free," improving the country's health.
Sunak also plans to bring forward measures to restrict the availability of vapes to children.
The government will consult on restricting the flavours and descriptions of vapes so that they can no longer be targeted at children, the briefing paper said, adding the government would also look at regulating vape packaging and presentation.
The proposed smoking ban is similar to one brought in by New Zealand last year, which became the first country to stop those born during or after 2009 from legally buying cigarettes.
The ban would take effect in 2027.
Sunak said smoking costs the UK's health services 17 billion pounds ($A33 billion) a year and cancer deaths could fall by a quarter if people stopped smoking.
The policy would hurt companies that earn a relatively large portion of earnings from their UK cigarette businesses, namely Japan Tobacco, maker of Camel and Benson & Hedges, and Imperial Brands, which produces Winston cigarettes and Golden Virginia rolling tobacco.
Campaign group Action on Smoking and Health welcomed Sunak's plans, adding they could hasten the day when smoking was obsolete.
The tobacco industry criticised the proposals.
The Tobacco Manufacturers Association said they were a "disproportionate attack" on adults' rights and would fuel black market trade.
"The prohibition of legal products always has dangerous side effects and opens the door to criminal gangs to sell illegal products," it said.
Imperial Brands also warned the ban threatened "unintended consequences".
Lucky Strike and Dunhill maker British American Tobacco said the proposals would be difficult to enforce.
The smoking policy would need to pass a free vote in the UK parliament.
This means MPs can vote however they like rather than in line with party policies.
