The San Francisco company, which boasts more than 50 million registered users, made the announcement via Twitter but did not disclose how much financial backing the musicians have provided.
"Dropbox is excited to welcome Bono and The Edge as investors. Thanks for the support and look forward to great things," read a tweet from the firm.
Bono's Elevation Partners has previously joined forces with other tech-based operations, including Facebook, Forbes, MarketShare, Move, Yelp and Palm. The singer reportedly makes close to $1 billion (£488 million) from his Facebook shares alone.
Dropbox is available across multiple operating systems including Windows, Mac OS X and Linux, as well as mobile platforms Android, Windows Phone 7, iPhone, iPad, WebOS, and BlackBerry.
The service is expected to face stiff competition from Google's upcoming Google Drive service in the near future.
Watch U2 perform 'Vertigo' live as part of their '360' tour below: