GFH has bought the club from majority shareholder Ken Bates, the former owner of Chelsea and a familiar figure in English football for the past three decades.
Bates, 81, is not severing all ties with Leeds. He will remain club chairman until the end of the season.
Well supported Championship clubs like Leeds are drawing interest from foreign investors who have their eye on promotion to the lucrative Premier League.
"Leeds United is a great football club, It's got history, it's got pedigree, it's got a fantastic base on which we can build. We looked at a number of clubs but for us Leeds United was the most attractive," GFH Capital executive Salem Patel told a news conference.
"We're not going to be spending crazy money like some football club owners have. What we want to do is to make the investment sustainable and make the club successful," he added, saying the business model was not predicated on the club being in the Premier League.
Money from Abu Dhabi has helped to turn Manchester City into Premier League champions but questions have been raised about the financial stability of GFH Capital and its Bahrain parent firm Gulf Finance House.
David Haigh, a GFH Capital executive and Leeds fan who has acted as the frontman for the bid, defended the takeover and the credentials of the company.
"We've bought this club with cash, there is no debt," he said. "We are a Dubai-based regulated bank, owned by a Bahrain-based regulated bank that is listed on four stock exchanges including London."
Leeds have not been in the Premier League since 2004 and are currently mid-table in the Championship. Their limitations were exposed when Chelsea beat them 5-1 in the League Cup this week.
The club last won the English top-flight title in 1992, the final season before the Premier League was launched.