The Committee on Climate Change (CCC) confirms that bills and emissions have been forced down since 2008 by EU energy efficiency rules.
Appliances like fridges, freezers and boilers are now designed to use less energy for the same work.
Ministers say they will make new energy efficiency policy in the Autumn.
But the government's critics point out that its Clean Growth plan for an efficient low-carbon society is already many months behind schedule.
The CCC says the UK must shift much more swiftly towards electric cars to reduce carbon and tackle local air pollution.
It also says a strategy is urgently needed to insulate millions of homes and create new forms of heating that don't foul the air or crank up climate change.
The report's finding on home energy bills will surprise people amidst allegations of energy company profiteering and fears that many households find energy prices too high.
It confirms that although the cost per unit of electricity and gas has indeed risen, household bills have fallen thanks to EU and UK efficiency standards which forced engineers to design appliances that use less energy.
It says since 2008, when the Climate Change Act was introduced, electricity demand is down 17% (despite all our gadgets) and gas demand is 23% lower, thanks to better insulation and UK rules on improved boilers.
The CCC chair Lord Deben told BBC News: "Really good EU regulations have made all our appliances much more efficient. It's meant we've been able to cut bills by about Â£20 a month for the average consumer.
"It's been a remarkable success story. But the crucial thing is we've got to keep on doing it."
It is not yet clear how the UK will be affected by future EU efficiency rules, although it will be impossible for UK manufacturers to sell inefficient goods into the EU market.
Catherine Mitchell, professor of energy at Exeter University, told BBC News: "I don't think most people have any idea about the positive role EU standards play in our appliances in our houses - and have done for the last 25 years, or so.
"They have brought down bills considerably, they have brought down carbon emissions and they have stimulated innovation. It would be quite mad to get rid of them when we leave the EU."
Meanwhile, the CCC says the government's carbon cutting policies are already off track to meet its obligation under the Climate Change Act to cut emissions of 57% by 2030 compared with 1990 levels.
The CCC wants: more low-carbon electricity through the 2020s; faster uptake of electric vehicles; a strategy for deploying the carbon capture and storage technology (still very expensive); and a strategy for the uptake of low-carbon heat, perhaps by switching from natural gas to hydrogen for home heating.
The new climate change minister Claire Perry told Parliament that she wanted to see if the government's embryonic Clean Growth plan could be made more ambitious. She promised to bring forward policies in the Autumn.
But green groups question the government's sincerity. They say that in a meeting of EU energy ministers, France, Germany, Portugal, Sweden, Italy, Luxembourg and Republic of Ireland supported improvements in energy saving of 1.5% per year.
But the UK, backed by Malta, Poland and others, wanted flexibilities and lower, non-binding targets.
Imke LÃ¼bbeke from the green group WWF said: "It should have been a no-brainer to support strong energy efficiency measures given the economic, climate and social benefits of doing so, and the ambition required by the Paris Agreement.
"Despite this, member states have rushed to appease the lowest common denominator to reach a quick but bad deal, rather than putting in more time to try and get a better deal."
A government spokesman told BBC News: "The government is a world leader in tackling climate change and committed to meeting the UK's targets set under the Climate Change Act 2008.
"We have cut emissions by more than a third while growing the economy by over two thirds, and continue to support our burgeoning low-carbon sector."
One difficult challenge for the government's Clean Growth plan will be to persuade home owners to insulate their properties better, following the failure of the Green Deal finance scheme.
Carbon and energy price targets can't be achieved unless insulation is improved, but the public is loathe to suffer the disruption and cost of works which will take years to pay back, and ministers baulk at throwing grants at householders to improve the value of their properties.
One campaigner, Simon Roberts from the Centre for Sustainable Energy, offers a glimmer of a solution. He suggests an incremental approach in which the government and local councils work to develop skills in a home insulation industry by helping people who have spare cash to invest in measures that improve their comfort as they age.
He told BBC News: "We've got ourselves paralysed trying to get the whole housing stock up to standard rather than thinking about how we get the next million done.
"There are people willing to invest in retro-fitting their houses and builders ready to do it. If we can find ways to orchestrate that at a local level you could start to change the whole dynamic of how our houses get retro-fitted and create a new social norm."
The climate sceptic group Global Warming Policy Forum (GWPF) urges the government to consider the social effects of allowing energy costs to rise in the assumption that they will be offset by energy savings.
Its spokesman John Constable told BBC News: "All consumers will be seeing the price increases that result from climate policies, but not all consumers will benefit from the energy efficiency measures since these typically require the purchase of new appliances. That means the consumers who benefit tend to be better off."
Energy isn't the only issue tackled in the CCC report. It also focuses on adaptation to changes already underway in the climate.
It warns that key infrastructure in the UK is still vulnerable to the sort of extreme weather events expected as the world warms.
The CCC admits that some action has been taken to address the risks from warming, like funding for flood defences. But it says much more needs to be done to make Britain's infrastructure resilient to extreme events.
The government spokesman said: ""We've implemented the majority of recommendations from the first National Adaptation Programme report and are pleased the CCC recognises the progress that has been made, but we agree there is a need to do more."
Friends of the Earth climate campaigner Simon Bullock said: "This is the starkest warning yet from the climate committee that the UK government must up its game.
"Climate change threatens the security of everyone at home and abroad and the continual delay from ministers on producing a fit-for-purpose strategy is simply unacceptable. "
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Household energy bills and carbon emissions will rise unless ministers devise new policies to save power, a report says.