The 23-year-old striker was on target twice in the second half to help the Dons ease past Graham Turner's men, with Byron Harrison opening the scoring two minutes after the break.
Shrewsbury did manage one goal back through Australian Shane Cansdell-Sherriff but brought the curtain down on their campaign with a defeat.
The visitors dominated the first half and should have scored when Terry Gornell was sent clean through, but Seb Brown saved well at his feet.
Immediately after half-time the Dons broke through, Harrison slotting home after keeper Chris Neal fumbled Billy Knott's cross. Jack Midson pulled the ball back for Moore eight minutes later and the young striker doubled the lead with a half-volley before Cansdell-Sherriff got one back in a goalmouth scramble just before the hour.
However, Wimbledon were not to be denied and sealed all three points after 61 minutes as Moore latched on to an excellent through-ball from Knott.