WASHINGTON—President-elect Donald Trump on Wednesday turned to a third retired military officer to help him run the country when he takes office in January, a move that represents an unusual level of military influence in the executive branch.

Mr. Trump gave the nod to retired Marine Gen. John Kelly to lead the Department of Homeland Security, according to people familiar with the decision, a nomination that comes after he already has asked retired Marine Gen. James Mattis to be his defense secretary and retired Army Lt. Gen....