Trustees of the BLF approved the move yesterday, a week after Asthma UK's members agreed to the same course of action at an extraordinary general meeting.
The charities, whose combined income is about £17m a year, will continue to operate publicly using their old names and brands for the foreseeable future, but will begin working "in parallel" on a joint strategy and business plan in the new year, a spokesman said.
The merger is expected to save £2m on running costs, which will be spent on research.
But it could lead to the loss of up to 20 jobs.
A BLF spokesman said discussions on redundancies would start in January and it was hoped that natural turnover of staff meant fewer than 20 people would be affected.
He added that it was too early to say which departments would suffer job losses.
Kay Boycott, chief executive of Asthma UK, who has been appointed chief executive of the merged organisation, said: "This marks the start of a new era that promises to bring significant benefits to people affected by asthma and other lung diseases.
"Bringing together our joint vision and strategy will allow us to make even more impact, with more money to spend on ground-breaking research and support.
"Our new organisation will combine the energy and passion of the UK’s top respiratory health charities, creating a powerful voice for change."
Both organisations, which announced plans to merge in November, will continue to carry out their core charitable activities. These include funding research programmes, campaigning and running helplines.
Asthma UK, which employs 84 staff, had an annual income of £7.8m in the year to 30 September 2018, its latest accounts show.
The BLF, which employs about 100 staff, had an income of £9.6m in the year to 30 June 2018.
The spokesman added that the new organisation had yet to identify a role for Penny Woods, the existing BLF chief executive, who is due to stay.
Baroness Tessa Blackstone, chair of the BLF, will chair the new board, which will have an equal number of trustees from both organisations.
Both charities are based in London.
The spokesman said the BLF's offices, which the charity owns, were too small to accommodate the merged organisation, so it was likely to be based in Asthma UK's premises in Mansell Street, although talks were ongoing.