He replaces Barrie Hubbard, who has chosen to step down after 29 years on the club's board.
Allen, who had previously held the position of chief executive, told the club website: "I intend to give 100% in my efforts to achieve what the fans want, which is Championship football."
Meanwhile, midfielders Danny Whitaker, 31, and Mark Randall, 18, have agreed new one-year deals.
Whitaker, who joined the Spireites from Oldham in 2010, made 35 appearances in 2011/12.
Randall played 18 games in his first season with the club, having joined them from Arsenal last summer.
Chesterfield have appointed Dave Allen as their new chairman.