While even small amounts increase the risk, drinking above recommended limits causes the majority of cancer cases linked to alcohol, experts said.
Even former drinkers who have now quit are still at risk of cancer, including cancer of the oesophagus, breast, mouth and bowel.
NHS guidelines are that men should drink no more than three to four units a day while women should not go over two to three units a day.
But yesterday’s research, published in the British Medical Journal (BMJ), found cancer risks at even lower levels.
Experts analysed data from eight European countries, including the UK, and worked out what proportion of men and women were drinking above guidelines of 24g of alcohol a day for men and 12g a day for women. In the UK, one unit is defined as eight grams of alcohol, meaning 12g is roughly a small 125ml glass of white wine (1.6 units).
A 175ml glass of red wine at 13 per cent strength is the equivalent of 18.4g of alcohol (2.3 units), as is one pint of 4 per cent lager. A double shot of whisky is 16g of alcohol (two units) while a 175ml glass of champagne is 16.8g (2.1 units).
The research found that men in Germany were the most likely to exceed 24g a day (43.8 per cent of the male population), followed by Denmark (43.6 per cent) and the UK (41.1 per cent).
Among women, Germans were most likely to drink over 12g a day (43.5 per cent of women), followed by those in Denmark (41 per cent) and the UK (37.7 per cent).
Cancers of the pharynx, oesophagus and voice box were most commonly caused by alcohol, followed by cancer of the liver.
Overall, three per cent of cancers in men were linked to drinking less than 24g of alcohol a day but more than 18 per cent were down to drinking more than 24g a day.
In women, one per cent of cancers were down to drinking less than 12g of alcohol a day while 4 per cent were due to drinking more than 12g of alcohol daily.
Some 17 per cent of bowel cancers in men were linked to drinking as were four per cent of cases in women.
And five per cent of breast cancers in women were also down to drinking, the study showed.
Even more cancers were thought to be partly attributable to drinking, and for every additional drink a day, the risks went up.
The authors, from universities and hospitals across Europe, said: “Our data show that many cancer cases could have been avoided if alcohol consumption is limited to two alcoholic drinks per day in men and one alcoholic drink per day in women, which are the recommendations of many health organisations.
“And even more cancer cases would be prevented if people reduced their alcohol intake to below recommended guidelines or stopped drinking alcohol at all.”
Cancer Research UK, which helped fund the study, said the findings showed that alcohol causes at least 13,000 cases of cancer a year in the UK.
More than 6,000 of cancers of the mouth, oesophagus, voice box and pharynx are caused by drinking.
Alcohol also causes more than 3,000 bowel cancers and about 2,500 breast cancers per year, according to the charity.
The research is part of the European Prospective Investigation of Cancer (Epic), one of the largest-ever studies into the links between diet and cancer.
It involved more than 360,000 men and women who were mostly aged 35 to 70 at the start of the study.
Their alcohol intake at recruitment stage in the 1990s was noted, and the follow-up examined how many developed cancer.
Alcohol creates a chemical when it is broken down by the body which can damage DNA and increase the risk of cancer.
Naomi Allen, from Oxford University, who works on the Epic study, said: “This research supports existing evidence that alcohol causes cancer and that the risk increases even with drinking moderate amounts.
“The results from this study reflect the impact of people’s drinking habits about 10 years ago.
“People are drinking even more now than then and this could lead to more people developing cancer because of alcohol in the future.”
The director of health information at Cancer Research UK, Sara Hiom, said: “Many people just don’t know that drinking alcohol can increase their cancer risk.”