George Friend headed Exeter into the lead but Reid ensured the Addicks a share of the spoils when his shot was adjudged to have crossed the line.
The first half was an even affair with defences on top. Exeter's best chance came when a good move down the left ended with Friend crossing for Richard Logan and he laid the ball off to Barry Corr. The striker shot for goal but Darren Randolph got down well to save.
Frazer Richardson's free-kick was well saved by Paul Jones before Exeter broke the deadlock after 59 minutes. Ryan Harley crossed from the left and Rob Edwards flicked the ball on at the near post before Friend headed in from close range.
However, Charlton equalised in the 72nd minute when Nicky Forster crossed from the left and Reid turned the ball goalwards. Scott Golbourne hacked the ball clear but the linesman flagged that the ball had already crossed the line.
Both sides had chances to win it in a frantic finale.
Harley came closest but his shot struck the upright while Randolph and Jones both made vital saves.