The Shrimpers edged closer to an unlikely play-off place thanks to Barnard's second-half penalty.
Robinson though was guilty of a shocking miss when he fired wide of an open goal just after the re-start.
That could have proved as costly as it was embarrassing but Barnard made sure the blunder didn't matter.
Neither side managed a single shot on target in a dismal first half and it was Southend who looked most dangerous.
Martin Kelly cleared from Damian Scannell and a cross from Franck Moussa skimmed off Robinson's head.
Andy Butler blocked Scannell's goalbound effort and Simon Francis drilled a free-kick into the wall before Alan McCormack squandered a great chance after a one-two with Robinson.
Robinson, with five goals in his previous nine matches, should have scored four minutes into the second half.
The striker latched onto a long clearance and keeper Alex Smithies came to meet him but missed the ball.
Robinson turned Andy Butler and had time to place his finish into the gaping net but right-back Tom Clarke did enough to force him to prod wide.
To howls from the Huddersfield fans, Robinson pulled his shirt over his head.
Barnard made amends in the 63rd minute when he grabbed the winner from the penalty spot.
Moussa's driven cross struck Tom Clarke's arm and Barnard made no mistake for his ninth of the season and fourth in six games.
Robinson was denied by Smithies after a Huddersfield defensive mix-up but the home side finished the strongest.
Butler headed just wide and Andy Booth also missed the target before Gary Roberts saw his shot kept out by Steve Mildenhall in stoppage time.