The deal, which heralded a "new era" for the club, had been approved and ratified by the Premier League, and the firm's Managing Partner Alan Pace would replace Mike Garlick as club chairman, it added.
Garlick and former shareholder John Banaszkiewicz will stay on ALK Capital's new board of directors for the club.
"Today marks a new era for Burnley as we become stewards of this historic club and build on the impressive work that Mike Garlick, (manager) Sean Dyche and everyone at Burnley has done to make it a financially stable, established Premier League club that is a cornerstone of the local community," Pace said.
"With a rich heritage, a brilliant academy, and a passionate fan base, this club has solid foundations to build upon. This is the start of an exciting journey for the entire Clarets family."
Reuters had reported in September that ALK were in talks to buy the club and Burnley confirmed in November that they were in discussions with "interested parties".
Pace has a Wall Street background, having held a senior role with Citigroup after previously working as a managing director for Lehman Brothers.
He was president of Major League Soccer club Real Salt Lake and has recently brought football scouting and recruitment technology products to the British market.
Dyche's relationship with Garlick had been strained over the past year with the manager unhappy at the failure to make significant additions to his squad.
Burnley beat Sheffield United 1-0 on Tuesday to move up 16th place and climb five points above the relegation zone.
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