WASHINGTON—The Justice Department has decided to allow Bayer AG’s megadeal to acquire Monsanto Co., valued at more than $60 billion, after the companies pledged to sell off additional assets to secure government antitrust approval, according to people familiar with the matter.

An agreement in principle between the companies and the department, brokered in recent days, marked a breakthrough in the U.S. merger review process, which had remained in limbo because of Justice Department concerns about the deal.

Monsanto...