The striker slotted home from the spot after 63 minutes after Huddersfield's Tom Clarke was penalised for handling Franck Moussa's cross
Southend edged a lacklustre first half with Damian Scannell and Alan McCormack wasting their best chances as the duo both shot over the bar.
Huddersfield came close to opening the scoring after 48 minutes when Anthony Pilkington turned neatly in the area but his crisp drive was tipped around a post by Steve Mildenhall.
Robinson scuffed his effort off target soon after as Southend missed an opportunity to move in front before Barnard spared his blushes with the winner from the spot.
Huddersfield twice wasted chances late on as Andy Butler headed off target before Gary Roberts' fierce volley was almost turned home by substitute Andy Booth.