The 29-year-old striker struck for the second game running after 72 minutes to take him to 12 goals for the season and drop Aldershot into the relegation places.
The first chance arrived after 11 minutes when Lionel Ainsworth's cross-shot was well saved by John Sullivan and Craig Reid was unable to force the ball in from four yards.
After long periods of scrappy play, Ainsworth had another effort just before the hour mark but Sullivan tipped his shot onto the post.
Soon after Sullivan was called into action again, this time denying substitute Jeff Goulding from outside the area.
Aldershot controlled the game for long spells but the Dons took the lead with less than 20 minutes to go.
Poor defending from the Shots allowed the ball to fall to Midson who beautifully lofted the ball over Jamie Young and into the net to win it.