Maguire was out of contract at Pittodrie, but the SPL club were entitled to compensation for the 22-year-old and had spent several weeks discussing a fee.
The Scotland international had turned down the offer of a new contract with Aberdeen in favour of testing himself in the Championship next season.
He has signed a three-year contract at Pride Park and he becomes Rams manager Nigel Clough's seventh capture of the summer.
"Chris is an exciting talent who has great potential to improve further," Clough told the club's official website.
"He is a creative player and someone who can bring a spark to the team.
"He's still young and will develop and improve, but having played in the Scottish Premier League and made his international debut he already has the quality to be competitive in our squad and team."