Moore’s 15th goal of the season and an assist for Mamadou Thiam earned the visitors all three points after Joe Pigott had cancelled out Cauley Woodrow’s opener, with Cameron McGeehan adding some late gloss with the Reds’ fourth.
Barnsley took the lead when a corner found Woodrow, who finished with aplomb on the turn with a first-time strike.
Wimbledon entered the game with the division’s worst attack but breached the league’s meanest defence as Pigott flicked home Andy Barcham’s free-kick.
The hosts should have led at the break but Jake Jervis missed a tap-in at the back post, while Dons keeper Aaron Ramsdale denied Kenneth Dougall and Alex Mowatt with sharp saves at the other end.
Jervis’ miss proved costly when Moore restored the visitors’ lead with a towering header from Mowatt’s cross.
Thiam then made it 3-1 with a rocket into the top corner after Moore had beaten two Wimbledon defenders to set him up before McGeehan compounded the Dons’ misery late on.