The two clubs have agreed an undisclosed fee for the Northern Irishman, reported to be around £800,000.
The midfielder, who has signed a contract until 2016, originally agreed to join Saints earlier this month but a question mark hung over the deal, with Ibrox chief executive Charles Green challenging the legal right of several Rangers players not to transfer their contracts to his newco club.
"Saints' executive chairman Nicola Cortese has now reached a favourable agreement with Rangers that is in the best and fairest interests of all three parties.
"This negates any possible delay in the transfer proceedings and sees Davis' registration transferred to Southampton with immediate effect.
"As part of this alteration to the transfer, Davis' contract with the club will now run until 2016."