The striker took just five minutes to open the scoring. A poor Darren Holden clearance only went as far as Will Atkinson on the edge of the penalty area and the midfielder broke forward to cross low andfind predator Eastwood, who clinically finished.
The visitors were down to ten men on half-time when, a minute after a melee involving 15 players only saw a single yellow card issued, Corr saw red.
He tapped the legs of Pools defender Jack Baldwin from behind and, on the advice of an assistant, was given his marching orders by referee Mark Brown.
Cooper's side pressed Southend back and peppered their penalty area in the second half but to no avail and Phil Brown's side saw the game out comfortably.