Barnard, who had not scored for five months, broke the deadlock from close range in the 23rd minute of Tuesday night's League One clash at the Withdean Stadium after goalkeeper Mikkel Andersen parried an effort by Osei Sankofa.
Striker Lloyd Owusu, on loan from Cheltenham and making his home debut for Brighton, levelled a minute before the break, heading in a cross from Gary Borrowdale.
Owusu was denied by the post shortly after the interval and Brighton dominated possession before Barnard hit a spectacular shot into the top corner five minutes from time.
Strike partner Robinson condemned Brighton to their eighth defeat in the last 10 home league games by netting from six yards a minute from time following good work by Kevin Betsy.
Robinson, well-placed in front of goal, headed wastefully over after winger Betsy had provided the opening. Robinson, fed by Anthony Grant, should have doubled Southend's lead moments before Davies burst through for Brighton but failed to test goalkeeper Steve Mildenhall.