The 23-year-old has been in discussions over the move all month having failed to work his way into the Hoops' plans, ending his time at Parkhead with just 31 appearances over two years.
Announcing the deal, Miami mistakenly referred to the Scottish champions as a "Scottish Championship side" but swiftly corrected the error on their official website.
"Lewis is an exciting player that will contribute to our attack," sporting director Paul McDonough said of the deal. "He is a pacey and dynamic player who operates well in space. He has shown his quality in Scotland and in England and we're happy to acquire him."
Morgan is the most high-profile signing to date for Miami, who will begin their first season of MLS football on March 1.