The 25-year-old, who was released by Rotherham United last week, will move to Hillsborough once his Millers contract expires on June 30.
A target of Peterborough United boss Barry Fry, Branston opted to stay in South Yorkshire and join The Owls, as manager Chris Turner looks to turn the club's fortunes around.
"Guy is the sort of centre half I feel we need₻We need steel at the back," Turner told the club's official website.
"He knows the Second Division very well."
The Owls, meanwhile, have continued their pursuit for new faces by increasing their bid for Luton Town striker Steve Howard.
Wednesday revealed that The Hatters turned down a second offer of £300,000, so they have decided to increase that bid to £500,000.
The club are determined to land a proven goalscorer next season, with Howard having netted 15 goals this past season for the Kenilworth Road club.
Another summer target is Blackpool striker Scott Taylor, but Seasiders chairman Karl Oyston has been quick to stress that no bid had been received from Wednesday for the hit-man.
The Bloomfield Road chief was surprised to see that The Owls had publicly declared a bid had been made for the goalscorer, having stressed that only an initial enquiry had been made for Taylor from their Second Division rivals.
"Sheffield Wednesday's chief executive contacted me yesterday (Wednesday) regarding Scott," Oyston told Blackpool's official website.
"They have expressed an interest and said they would follow it up with an offer, but no figures were discussed and no subsequent offer has been made.
"We are extremely surprised that they have chosen to go public and I would question their motives for doing so, perhaps they are trying to boost season ticket sales to pay off some of the £20 million plus debt they have."
