Naas Olivier scored three penalties for the home side who led 9-3 with just two minutes remaining but a penalty try awarded for a dominant scrum in the dying seconds broke their hearts.
The Dragons arrived in high spirits after their win over Edinburgh at Rodney Parade a week ago and took the lead after a quarter of an hour through a Jason Tovey penalty.
However in a scrappy opening half neither side was able to take control and, despite the sin-binning of Matteo Pratichetti, the Dragons could not add to their total.
In fact it was the home side who drew level just before the break after Anitelea Tuilagi had also been binned, Olivier slotting with the final kick of the half.
Robling replaced Tovey at half-time but it had little impact on the game with both sides struggling to break down the dominant defences.
Midway through the half Olivier finally got the next score with his second penalty as Aironi moved into the lead for the first time.
Three minutes later he added another three points to stretch the lead to six, and with neither side threatening a try it looked like it might be enough.
However after Italy hooker Fabio Ongaro was sent to the bin two minutes from time the Dragons took advantage of a weakened Aironi scrum to force the penalty try.
Robling, whose misfiring kicking cost the Dragons RaboDirect PRO12 victory against the Scarlets in December, held his nerve to slot the easy conversion and give the Dragons their first away win of the campaign.
The Welsh region now lie ninth in the RaboDirect PRO12 table after leapfrogging Edinburgh and Connacht ahead of next weekend's clash with reigning champions Munster.
|Aironi Rugby Score Card|
|Newport Gwent Dragons Score Card|