Deaths fell from about 580 per year to 300 in this age group while the largest drop was in those with leukaemia.
More specialised treatments are likely to be behind the trend, the report said.
However, a teenage cancer expert said more young people should be enrolled on clinical trials.
Cancer remains the main cause of death from any disease in teenagers and young adults. Only transport accidents account for more deaths in this age group.
The report, Cancer Statistics Report: Teenage and Young Adult Cancer, calculated that about 2,100 young people aged 15-24 years old are diagnosed with cancer each year in the UK.
But in the past 30 years or so, death rates have fallen in males from 88 deaths per million (in 1975-1977) to 44 deaths per million (in 2008-2010) and in females from 61 deaths per million to 31.
Leukaemia deaths among teenagers and young adults have seen the greatest drop since 1995 in the UK - from an average of 54 per year to 39 in 2006-2010 in young males and from 38 to 21 deaths per year among females.
Brain tumours were the most common cause of cancer deaths in this age group between 2008 and 2010.
"More investment in rare cancer research is urgently needed. We want to work with Cancer Research UK and the pharmaceutical industry to ensure better access to clinical trials for young people with cancer."
The report said that less than 20% of patients aged 15-24 with cancer take part in clinical trials, compared to 50-70% of child cancer patients under 15 in the UK.
It added that broadening access to clinical trials was essential to improve knowledge of the best treatments for cancers.
|Dr Harpal Kumar|
"Drug development and clinical trials are at the heart of helping more teenagers and young adults both survive cancer and live a full life after their treatment.
"Too many young people are left out of clinical trials due to rigid age restrictions and this must change for us to continue to see improvements across all cancer types."
Those restrictions exist because there are potential dangers in giving young cancer patients adult doses of drugs, therefore researchers are often reluctant to develop trials which cover this age group. Children's cancer drug trials are also very specialised.
Although cancer mortality rates for young people are falling, the incidence of all types of teenage and young adult cancers combined has been rising since the 1990s.
The report was a collaboration between the North West Cancer Intelligence Unit, on behalf of the National Cancer Intelligence Network, and the Paterson Institute of Cancer Studies, at the University of Manchester.