First Utility said remaining on more expensive tariffs with one of the Big Six companies has led to households losing an average of £800 each.
The firm compared how much people on the energy giants' standard variable tariffs would have spent over the period with how much they could have spent if they had switched to a cheaper supplier and tariff plan.
First said it used information from switching site Energy Helpline and a report by the Competition and Markets Authority (CMA).
The supplier provides gas and electricity services but is not one of the Big Six of British Gas, Npower, Scottish Power, EDF, E.On and SSE.
First Utility claimed many customers remained unaware they could save money "in part thanks to the Big Six energy suppliers doing their best to keep their most loyal customers in the dark about savings and alternative tariffs".
It said part of the problem came from people only receiving bills every three months.
Ed Kamm, UK managing director of First Utility, said: "The Big Six have been exploiting customers' loyalty for too long and it has to end.
"The brutal truth, hidden away in the CMA report, proves that the Big Six have been relying on their standard variable tariff customers for years to bolster their profits.
"We have to see real change in 2017, with the onus on helping those who have been kept on bad deals for years and years."
Most of the major suppliers have announced price rises in the last few months.
Later, MPs will put a motion on energy prices to the House of Commons in a debate in which a cross-party group will call for measures to control the market and protect consumers.
The Department for Business, Energy and Industrial Strategy has previously said the Government will act whenever markets are failing consumers.
It came as a report was released by a Government committee that said climate policies have not been responsible for increased bills.
A report by the Committee on Climate Change said the efficiencies that have resulted from the green policies have saved consumers, on average, as much as they have cost.
A Business and Energy Department spokesman said: "This report ... suggests recent price rises announced by energy suppliers cannot be explained by policy costs alone."