Corr fired into the roof of the net with 20 minutes remaining while Assombalonga won and converted a late penalty for the Shrimpers to seal a first victory in six games for former Hull manager Brown.
Aldershot, who are four points adrift of safety with just two games remaining, saw Peter Vincenti and Michael Rankine go close in the opening half hour.
Southend goalkeeper Dan Bentley was then tested eight minutes before the break by Paul McCallum's looping header, but the hosts faded after the break.
Assombalonga broke forward and hit the post with a weak effort before Corr snatched the opening goal after 70 minutes following a goalmouth scramble.
Eight minutes later, Assombalonga grabbed his 16th of the season after tripping under the challenge of Troy Brown inside the box.
Asa Hall dragged a left-foot shot wide for the Shots late on but Southend held firm for their first win in eight matches.