The League Two club received more than 100 applications for the post.
Edwards, a former Wolves Under-23 coach, will take up his new role on June 5 and will be assisted by Richie Kyle.
Kyle had worked alongside Edwards in the England youth set-up before accepting the post of assistant to the Canada Women’s team.
Edwards said: “This is a club that feels really different from others – and actually does things another way.
“My conversations with the chairman were incredibly positive, and showed we’ll get the time and support we need to go on a winning journey.
“The club has a well-developed footballing infrastructure under (football director) Richard Hughes, and with great people behind the scenes and incredible fans, I’m looking forward to getting started.”
The club received more than 100 applications for the post, but opted for former Aston Villa, Wolves and Blackpool defender Edwards in part for his proven track record in player development.
Chairman Dale Vince said: “We recently moved away from a manager to the head coach concept – with this recruitment we’re completing that transformation – by bringing in an up-and-coming coaching team.
“We wanted to move away from the recruitment merry-go-round of been there and done it league managers to focus on coaches at the cutting edge of the game.
“Rob and Richie bring an understanding and openness to all aspects of coaching – like sports science and analytics. This is the start of our next era.”
Rovers ended Mark Cooper’s five-year reign in April with their League Two promotion drive faltering and although they eventually made into to the play-offs with Under-18s boss Jimmy Ball at the helm, they were beaten 5-4 on aggregate by Newport in the semi-finals.