The Dons were in danger of sliding to a surprise defeat when Bowditch took aim from the edge of the penalty area to smash a volley past goalkeeper Chris Weale.
It was cruel on Shrewsbury, who were seconds away from claiming a notable scalp to boost their fight against escaping the relegation zone.
Bowditch was earlier the provider as the Dons moved ahead on the stroke of half-time. He broke with pace down the right and crossed for Angelo Balanta to fire home from 12 yards into the bottom corner of the net.
Shrewsbury produced a tremendous response in the second half to take the lead. Luke Rodgers, on loan from Portsmouth, grabbed the equaliser in the 64th minute after Marvin Morgan rounded goalkeeper David Martin before unselfishly setting up his strike partner.
Then winger Mark Wright completed the turnaround eight minutes later when he bent a fine shot in off the far post against one of his former clubs. But Dons kept going and Bowditch earned them some reward.