The U.S. Justice Department approved Walt Disney Co.’s proposed $71 billion acquisition of 21st Century Fox Inc. assets on Wednesday, on the condition that Disney divest Fox’s regional sports networks.

The approval gives Disney a leg up in its battle with Comcast Corp. for control of key pieces of Rupert Murdoch’s entertainment empire. The conditions placed by the Justice Department allow Disney to still absorb the parts of Fox that have become key to its long-term strategy, namely the company’s movie and television studio,...