Key, 49, becomes director of rugby at Headingley Carnegie with 39-year-old Back - a former Lions player and World Cup winner with England in 2003 - assuming the role of head coach.
The former Leicester pair, who have signed three-year contracts, succeed Stuart Lancaster, who has become the Rugby Football Union's head of elite player development.
Key said: "Neil and I are massively excited by the challenge ahead. We recognise the situation that the club finds itself in and we want to put Leeds Carnegie back in the Guinness Premiership next year.
"We both come from a background of success but we are not naïve to the challenges that face us in National Division One.
"However, as outsiders we can see that Leeds are an established club, who know what they want and, although they fell short last season, they have a squad and backroom staff who are hugely enthusiastic and committed.
"The backing the club receives from Leeds Metropolitan University provides a fantastic opportunity for any coach and that certainly encouraged us to take this opportunity.
"We are looking forward to the start of the season and we can see from last season that the club has real strength in its staff and playing squad that hopefully we can develop and bring our style to the club in the process."
Back, who captained England in four matches, added: "After meeting (chief executive) Gary Hetherington and seeing the set-up both at the club's training ground and at Leeds Met, I knew that Leeds Carnegie presented me with the ideal opportunity to further my coaching career.
"I have had some fantastic times at Leicester and it was a big decision to take myself out of the application process for the director of rugby position there and end my long association with the club.
"However, Andy and I are excited about coming to Leeds, starting on Monday, and getting down to work with the coaching staff and players to begin our preparations for the challenges ahead."
Leeds Carnegie chairman Professor Simon Lee said: "Since the sports minister's announcement on Thursday that the UK Centre for Coaching Excellence has been awarded to our university, with the task of establishing the number one coaching system in the world by the 2016 Olympics, this is the first opportunity we have had to support that declaration of intent.
"By appointing Andy and Neil together, Leeds Carnegie are demonstrating the commitment of Leeds Rugby and our university to coaching with a difference.
"Andy and Neil will establish a coaching system that is all-encompassing and a style that is distinctive."
Hetherington added: "It is the first time we have recruited our director of rugby from outside of the club since Phil Davies was appointed in 1996.
"Over the past 12 years, the club has changed remarkably and now the challenge for Andy and Neil is to take us forward to the next level.
"The first immediate aim is to return to the Guinness Premiership next season and both coaches will take charge of pre-season training on Monday to begin preparations for our National Division One campaign."