Davis signed a deal to join the Premier League newcomers earlier this month, after refusing to transfer his contract to the financially-stricken Scottish side's newco.
But the move was delayed when Rangers put in a claim for compensation.
Southampton have agreed a fee with Rangers, and Davis has moved with immediate effect.
Davis was already training with Southampton, who are in France on a pre-season training camp.
"I'm delighted to get everything sorted," said Davis.
"There was an element of uncertainty at how long it would take to get the transfer papers through, so I have to give credit to the Executive Chairman Nicola Cortese for wanting to get it done and removing the need for things to drag on any longer.
"This is good news for all parties, and now we can all move forward."