Lord Davies, Britain's former trade minister, said women now accounted for 20.7% of board members in the FTSE 100 firms, up from 12.5% in 2011.
He wants women to occupy a quarter of all board positions by 2015.
The report also said there was a "growing recognition" of the benefits of equality to business.
The social and economic benefits of having more women on boards were also increasingly appreciated, Lord Davies' review found.
He said: "The rate of change that we have seen at the heart of our biggest companies over the last three years has been impressive.
"The voluntary approach is working and companies have got the message that better balanced boards bring real business benefits.
"We are finally seeing a culture change taking place at the heart of British business."
But he said the "eyes of the world" were on Britain, and the UK needed to prove it could promote equality on a voluntary basis, not by regulation.
Businesses are coming round to having women on boards, Newton Investment Management chief executive Helena Morrissey told the BBC.
"Mysteriously, the chairmen often say that the men behave better when the women are around," she said.
"Really the arguments are that you need diversity of thought, diversity of perspective," she added. "This has moved away from being politically correct, this is a move to being regarded as good business sense."
Lord Davies published a review in 2011 called 'Women on Boards'. He said "real progress" had been made in the intervening years, with "more women than ever before" in top positions.
Business Secretary Vince Cable said the figures showed businesses were getting the right mix of talent "around their boardroom table".
He said: "98 of the FTSE 100 boards are now made up of at least one woman and we need fewer than 50 new women appointments to FTSE 100 boards to reach our target of 25% of women on all FTSE 100 boards in the next year.
"This is a huge improvement from where we started just three years ago."
Mr Cable called for a "renewed, concentrated effort" by company chairs and chief executives to "change the make up" of their "top table".
He added: "More needs to be done to improve the number of women in executive positions.
"These will be the CEOs of tomorrow and businesses still aren't tapping into the vast talent pool available to them."
Shadow minister for women and equalities Gloria De Piero said the UK was making "decent progress" but it was a "bit early to be celebrating" as women represented 50% of the population.
She said a lack of diversity led to poorer company performance and that Labour would consider introducing quotas if progress was not made.
Ms De Piero added: "But there's also a role for government to play in setting the right example to business.
"With just five women out of 33 in the cabinet, and women only making up 20% of government ministers, David Cameron and Nick Clegg have shut women's voices out of their own top table."
Maria Miller, minister for women and equalities, said: "The workplace was designed by men for men. Women do not need special treatment, they just need a modernised workplace that gives them a level playing field."
She said supporting women to fulfil their potential should be a "core business issue" for the "long term sustainability of our economy".
The boardrooms of top UK companies have undergone a "culture change", with a growing number of women in decision-making roles, a report has said.