Bankruptcy Judge Stacey Jernigan approved the sale at a Thursday court hearing, following a marathon auction that began the prior day and ended just after midnight.
The $593 million sale includes $385 million in cash and the assumption of $208 million of liabilities.
Greenberg and Ryan, the current president of the Rangers, outbid a rival group led by National Basketball Association team owner Mark Cuban and Houston freight-forwarding executive Jim Crane.
The Greenberg-Ryan group had Major League Baseball's backing. The final hearing on the team's bankruptcy plan was scheduled for Thursday.
Cuban's group had offered $390 million cash as part of a $598 million bid. But the Greenberg-Ryan bid was considered higher because of the way the bids were structured -- including a $10 million to $13 million breakup fee that would have been paid to the Greenberg-Ryan group if it had not won.
Cuban and Crane did not immediately return calls seeking comment.
The Rangers, currently fighting for a playoff spot, filed for bankruptcy protection in May in a move to close a sale to the Ryan-Greenberg group, which was picked as the preferred buyer in January.
The case is In re: Texas Rangers Baseball Partners, U.S. Bankruptcy Court, Northern District of Texas, No. 10-43400.
(Reporting by Sakthi Prasad in Bangalore, Ben Klayman in Detroit and Tom Hals in Wilmington; editing by John Wallace, Bernard Orr)