Scott Bennett fired the Grecians ahead at the start of the second half but Corr, who was making his comeback from 17 months out with injury, nodded Blues back level in the 78th minute against his former side.
And 10 minutes later Eastwood headed home at the back post to securee the points for Paul Sturrock's side.
Exeter were on top for long spells and went in front in the 51st minute when Matt Oakley's right wing cross was back healed at the near post by Pat Baldwin, enabling Bennett to sweep home right footed from six yards.
But Shrimpers boss Paul Sturrock responded by making a trio of substitutions and they helped to sway the match in the home side's favour.
Corr headed a deep right wing cross from Sean Clohessy in to the back of the net from close range and, two minutes from time, Eastwood nodded in a left wing centre from Chris Barker.