The business said that it would buy all the e-commerce and digital assets of the three brands, as well as their inventory.
The deal does not include the brands’ retail stores, concessions or franchises, putting their future as high street names at risk.
The deal comes after Asos bought other Arcadia brands, including Topshop and Topman, out of administration a week ago.
Boohoo chief executive John Lyttle said: “We are delighted to announce the acquisition of the assets associated with the online businesses of the three established brands Burton, Dorothy Perkins and Wallis.
“Acquiring these well-known brands in British fashion out of administration ensures their heritage is sustained, while our investment aims to transform them into brands that are fit for the current market environment.
“We have a successful track record of integrating British heritage fashion brands onto our proven multi-brand platform, and we are looking forward to bringing these brands on board.”
“We continue to grow our portfolio of brands and customer base, strengthening our position as a leader in global fashion e-commerce.”
In a separate statement, administrators from Deloitte confirmed that around 2,450 staff will lose their jobs at the brands’ remaining 214 stores, which will permanently close.
Staff have been emailed this morning and will be informed over the course of the day.
Approximately 260 jobs will be moving with the brands to Boohoo, mainly head office functions such as brand design, buying and merchandising, and the digital part of the business. Some other staff will also go through a transition period for a few months.
It means that all of Arcadia’s brands have been sold out of administration.
Administrators still have some property to sell from the business. They have raised £500 million to pay off creditors so far.