Chris Todd's side picked up a third National League victory in seven days courtesy of first-half strikes from Yemi Odubade and Jai Reason.
Odubade charged down goalkeeper Alan Julian's clearance, and saw the ball bounce straight into the net.
Reason then rounded off work from Joe Partington and Luke Coulson, which ended Bromley's six-match unbeaten run.
Eastleigh kept alive their hopes of a play-off place by beating mid-table Bromley for a fourth win in five games.