For people needing oxygen, the drug also helps cut the death rate by one fifth.
Dexamethasone has been tested as part of the Recovery trial, which is part of the largest clinical trial in the world testing existing medicines as treatments for the disease.
Scientists say that had they known about the drug's effectiveness at the start of the coronavirus outbreak, 4,000 to 5,000 deaths could have been prevented in the UK alone.
The Recovery trial has recruited more than 11,500 patients at 176 NHS hospitals around the UK.
The arm of the trial which investigated dexamethasone involved around 2,000 patients who received the drug and around 4,000 patients who did not.
Professor Peter Horby, from the University of Oxford, has been partly coordinating the research.
Speaking at a briefing for journalists on Tuesday morning, he said the drug should be rolled out across NHS hospitals within the next 24 hours.
In an interview with ITV News, he said: "Our jaws dropped when we saw the result. It's really a big result for a severe disease in a largely elderly population, to have that kind of impact on a cheap and widely available drug is really surprising."
He added: "We're really quite amazed, so it is a fantastic result. I very much doubt we will ever see better than this but what we may see is that we can really ramp up the survival rate by adding on additional interventions and drugs."
The Recovery trial is testing a range of existing medicines including drugs for treating HIV to see if they have benefit against Covid-19 patients.
It recently stopped giving patients the anti-malaria medicine hydroxychloroqiune after it was found to show no benefit.
Another arm of the Recovery trial is giving patients plasma taken from the blood of those who have recovered from Covid-19.
Antibodies contained in the plasma should help severely ill patients fight off their infection more quickly.
So far, experts have put much hope in trials of other drugs such as Remdesivir and hydroxychloroquine, but the success of the dexamethasone is the first time a drug has been shown to have an effect on mortality.
Initial results from a randomised trial of Remdesivir in the United States showed the drug reduces patients' time in hospital from 15 days to 11 days but it does not necessarily prevent deaths.
Government’s Chief Scientific Adviser, Sir Patrick Vallance, hailed the treatment as a "ground-breaking development in our fight against the disease".
He said: "This is tremendous news today from the Recovery trial showing that dexamethasone is the first drug to reduce mortality from Covid-19. It is particularly exciting as this is an inexpensive widely available medicine."
"This is a ground-breaking development in our fight against the disease, and the speed at which researchers have progressed finding an effective treatment is truly remarkable. It shows the importance of doing high quality clinical trials and basing decisions on the results of those trials," he added.
England's Chief Medical Officer, Chris Whitty said: "This is the most important trial result for Covid-19 so far. Significant reduction in mortality in those requiring oxygen or ventilation from a widely available, safe and well known drug."
He adds: "Many thanks to those who took part and made it happen. It will save lives around the world."
Professor Jonathan Van-Tam, the Deputy Chief Medical Officer for England said: "The positive findings on dexamethasone follow the disappointing findings on hydroxychloroquine.
"Together these two results illustrate the power of properly conducted clinical trials and the inherent danger of assuming things work without robust data."
He added: "Whilst tempting to do otherwise, it is always better to wait for the evidence. On the dexamethasone findings, this is very encouraging because the signal on reduced mortality applies to many of the patients admitted to hospitals and the drug is comparatively low priced and available worldwide."
A version of the drug that can be injected or taken orally was added today to a government list of medicines that cannot be exported from the UK. A tablet version of the drug had been on the list since April.
A government spokesperson said the move was made to "help ensure the uninterrupted supply for NHS patients."
They added: "If medicines in the UK may be needed by our patients they should not be diverted to other countries for financial gain.”
Coronavirus: Everything you need to know