The 25-year-old, who scored 186 points in 20 league appearances for 'Quins this past season but crucially failed to help them maintain their Premiership status, was called up to the Irish squad to travel to Japan on Saturday due to Paddy Wallace's hand injury.
The Limerick man, capped once at international level against Samoa in 2001, will have to fight hard to displace Alex King (fly-half) and Mark Van Gisbergen (full-back) in the Wasps line-up.
Staunton commented: "It's a great opportunity and I'm excited at the thought of joining what is currently the most successful club rugby team in England.
"I'm looking forward to working with Shaun Edwards and Ian McGeechan and it will be a great chance for me to learn from such a successful and experienced coaching staff.
Staunton's signature marks the second Irish one at the Causeway Stadium in the space of two months, after ex-Munster scrum-half Eoin Reddan was confirmed as a Wasps acquisition in early April.
Reddan, 24, is a direct replacement for Peter Richards, who has penned a deal with Gloucester, but the former Ireland Under-21 has the ability to challenge England's Matt Dawson for the first choice number 9 shirt.
The announcement of Wasps' new Irish signings coincided with the club's new sponsorship deal with Irish cider company Magners.
Wasps, whose new director of rugby McGeechan takes over the reins from departed former Ireland coach Warren Gatland in mid-July, are to have "Magners" emblazoned across the chest and collar of their new kit.