In a control group, which did not have the screening, far fewer of the cancers were discovered when they were still small and more had spread to the lymphatic system.
Prof Gareth Evans, the lead author of the study, said the trial demonstrates that annual scans are effective in detecting tumours earlier for this younger age group.
He said overdiagnosis – where people are treated for cancers that are unlikely to prove harmful – was “far less likely” to be an issue with this younger age group.
“For women with a family history, removing a non-invasive tumour so early in their lives is likely to be a cancer preventive,” Professor Evans said.
The study’s authors said that more analysis was needed on the risks, costs and benefits of extending the screening programme.
But Baroness Morgan called for the government’s forthcoming review of NHS screening programmes in England to consider the introduction of scans for women aged 35 to 39 with a family history of breast cancer.
Lives cut ‘heartbreakingly short’
If annual mammograms for this group of women were made widely available across all four of the UK’s NHS services, it could affect up to 86,000 women, the researchers said.
Breast cancer is the most common cancer in the UK, with about 55,000 women being diagnosed each year and 11,500 dying from the disease.
Between 5% and 15% of breast cancers are linked to a family history of the illness.
“We’ve long known that a family history can define a woman’s risk, and that breast cancer can be more aggressive in younger women,” said Baroness Morgan.
“So if we can intervene earlier for those at higher risk through annual screening, we believe we may be able to stop the disease cutting so many women’s lives so heartbreakingly short.”
An NHS England spokeswoman said possible changes to the screening programme will be considered in the review.
She said: “Breast cancer survival is at its highest ever and with improved screening a key focus of the NHS long-term plan, even more cancers will be diagnosed earlier.”