The 29-year-old was the joint-top scorer in the division last season and opened his account for the new campaign in the 35th minute, sweeping home after collecting Angus MacDonald's lofted through ball.
The visitors, relegated from League One last season, made much of the early running with Drew Talbot immediately involved, finding Marc Richards who teed himself up before volleying wide of Seb Brown's goal.
Talbot could have twice put John Sheridan's side ahead after that but found Brown equal to his stinging drives, before Warren Cummings cleared Jordan Bowery's goal-bound effort off the line.
Midson then made Chesterfield pay for their wastefulness in front of goal but the Spireites piled on the pressure in the second half in search of an equaliser.
Brown had to be at his best to deny Sam Togwell, diving low to turn his strike round the post but that was the best opportunity they could muster as the Dons held on.