Fleetwood boss Joey Barton was not in the dugout after receiving a two-match touchline ban and a £2,000 fine.
The Town boss had been charged with misconduct for the language he used towards referee Brett Huxtable after James Clarke netted the winner in injury time during their 2-1 defeat at Bristol Rovers.
And Barton's side conceded in the 16th minute as Andy Barcham blasted Mitchell Pinnock's left-wing cross past Alex Cairns.
Fleetwood had chances to level before the break as Ched Evans was thwarted by Dons keeper Aaron Ramsdale and Ash Hunter fired over from six yards out.
Fleetwood dominated at the start of the second half but AFC Wimbledon increased their lead against the run of play just 10 minutes in.
Anthony Hartigan's effort from distance just crept into the bottom right corner.
But the game was turned on its head in the 70th minute as Fleetwood netted twice in two minutes.
Paddy Madden bundled home Ross Wallace's set-piece and then Town won a penalty.
Evans was fouled in the box and the forward was the calmest man in Highbury as he coolly slotted home the subsequent spot-kick in the 72nd minute.
But the Dons regained their lead in the dying stages as Appiah drilled the ball home in the 90th minute.