Staniforth, brother of Wallaby winger Scott Staniforth, has represented Australia Sevens on the IRB circuit.
His senior club career started in 1999 with Australian Universities and Southern Districts before moving onto the New South Wales Waratahs Academy and B side as well as representing Australian Legends in the Hong Kong 10s.
With a reputation for powerful, pacey running and a long accurate kicking game, he moved to play for New Zealand NPC team Southland in 2003 before moving to Exeter Chiefs last season. He was instrumental in Exeter's drive for promotion to the Guinness Premiership but they were edged out by Bristol.
As his wife is a UK passport holder, Graydon will qualify as a European player when playing for Glasgow.
Glasgow Head coach Hugh Campbell said: "We have a fair number of young backs in the squad at the moment and we felt a player of Graydon's experience would have a lot to offer the squad as a whole both on and off the pitch. I'm sure his strong and hard running style will make sure he fits in well at Hughenden and we're very pleased to welcome him on board."
Staniforth added: "I just arrived with the squad this week. We've been at a training camp at RM Condor in Arbroath so it's been really enjoyable and a great way to get stuck in and meet everyone. Looking to the rest of the season I'll hopefully be able to contribute wherever I can and get a few games under my belt and hopefully a few wins too."