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The announcement, made in a speech to the Royal College of Nursing's annual conference in Harrogate, could spell the end of the British Formula One Grand Prix.
Other sports like rugby, snooker, darts, cricket and ice hockey could also lose around £10 million in sports sponsorship in Britain.
Mr Dobson told the conference: "We will ban tobacco advertising. It will cover all forms, including sponsorship."
But he acknowledged the move would come as a severe blow to sports organisers of major events like the Silk Cut Challenge Cup snooker final and the Embassy World Professional Darts Championship.
"We recognise that some sports, like some smokers, are heavily dependent on tobacco sponsorship. We will therefore give them time and help to reduce their dependency on the weed," he said.
In an uncompromising speech he said: "The tobacco industry kills around 120,000 of its customers every year. So it has to recruit 120,000 new smokers to its ranks each year to make up for the casualties."
He said the final details would have to be ironed out before a draft White Paper was published in the coming months.
Health professionals are delighted by the move, promised during the recent election campaign that saw Labour to a resounding victory.
In a statement, the Imperial Cancer Research Fund said: "We welcome any moves to ban tobacco advertising and sponsorship."
A spokesperson for the British Medical Association added: "It's unacceptable for tobacco, the major cause of preventable ill health, to be linked with sport."
And a representative from anti-smoking group ASH said: "We're delighted and excited. We hope to be able to help the Government to work out a draft Bill."
The tobacco industry is now seeking urgent talks with the government to discuss the issue.
In November 1997, the government decided Formula One racing would be exempt from the sponsorship ban. But weeks later, the press revealed that the prime minister, Tony Blair, had met representatives of Formula One before the decision was announced.
It also emerged that Formula One boss Bernie Ecclestone had donated £1 million to the Labour Party before the general election.
Although Labour and Mr Ecclestone strongly denied any connection between the donation and the exemption, Labour gave him back his £1 million.
In December 1997, the European Union passed a law banning tobacco advertising and sponsorship - bar Formula one - across Europe.
It was overturned by the European Court of Justice after appeals from the tobacco industry.
In The UK tobacco advertising in shops and newsagents was outlawed in December 1999 and tobacco sponsorship of sports ended in 2003.
In July 2005 the European Union imposed a ban on tobacco advertising and sponsorship of sporting events, apart from events that are purely local.
After the British Grand Prix advertising ban came into effect, some tobacco producers switched to using logos or colour schemes similar to their usual adverts to promote their products at races.
This was a severe financial blow to snooker which had to look for new backing. The 2006 snooker World Championship was the first in 30 years not to be sponsored by Embassy cigarettes.
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