Twice Reid put the visitors ahead but twice Moore responded - handing the Dons a valuable point in their battle against the drop while Exeter remain in eighth spot, just outside the play-off places.
Jamie Cureton had the game's first chance after 12 minutes but his curling effort from just inside the penalty area was tipped away by John Sullivan.
Two minutes later Moore elected to tee up Sammy Moore, with the goal gapping after an Artur Krysiak error, but his shot from distance was saved as Krysiak recovered in time.
Exeter took the lead when Reid was given time outside the box to drive a low shot into the corner past Sullivan after 36 minutes.
Luke Moore levelled just before the hour mark when he pounced after Exeter failed to clear but Reid duly bagged his second from a tight angle to put the visitors back in front soon after.
But on 63 minutes Luke Moore levelled with another low finish from outside the area, before substitute Charlie Strutton missed a one-on-one for the hosts and Cureton had a late goal disallowed for offside.