West Brom have sacked Nicolas Anelka citing "gross misconduct" after the striker claimed on Twitter he had quit the club.
Mauricio Pochettino expects to be at Southampton next season and is looking forward to working with chairman Ralph Krueger.
The 30-year-old defender will bolster Graham Turner's defence after Jermaine Grandison was ruled out for six weeks with a hip injury.
Turner was delighted to add Edwards to his squad ahead of the transfer window closing.
"To get some with Rob Edwards' experience is a big boost to the squad and we are delighted to have Rob in join the squad," he told the club's official website.
"With Jermaine being out of action for a while we needed to bolster our defensive options."
Edwards began his career at Aston Villa, before going on to enjoy spells at Wolves and Blackpool.
His last senior appearances came for Fleetwood on loan before Christmas, where he played five times.