The 22-year-old had been available on a free transfer this summer after leaving West Bromwich Albion at the end of last season.
Rose failed to make a single first-team appearance for the Baggies during his three years with the club, but has suitably impressed Southampton while on trial to earn a short-term deal at St Mary's.
"I came in for a week or so to get a feel for the place and show them what I could do and I'm excited to be here now. Everyone knows about the reputation Southampton have in bringing through young players and they give you opportunities," Rose told Southampton's official website.
"It makes it more real when you see the young players being given chances in the first team so that's one of my goals. There is a great pool of goalkeepers here who I can learn from to develop my game.
"Fraser [Forster] is an England international, Stuart Taylor has a lot of experience and I trained with Alex McCarthy in the week –he was really good with me. Dave Watson has been with England and is brilliant at what he does, so I can't wait to get going."
Rose has previously spent time on loan at Crawley Town and Accrington Stanley.