The Dragons crossed for two tries to one but were made to sweat on the win by the determined Italians, who were within a score of their hosts in the closing stages.
It is a defeat that leaves Aironi, who have 10 players on World Cup duty with Italy, still searching for their first away win since joining the Celtic League.
The Dragons have five Wales players in New Zealand themselves but they coped with their absentees better than the visitors to get their campaign up and running.
Both teams headed into the game on the back of two defeats but it was the Dragons, playing in front of their new £4million stand for the first time, who started more brightly.
They were straight on the front foot and got their reward when the ball was spread to the right after a series of drives for centre Tom Riley to cross over with just four minutes gone with Jason Tovey expertly adding the conversion.
That sparked the visitors into life but they were unable to get on the scoreboard despite a period of pressure on the Dragons' line.
That failure was punished when Tovey stretched the lead with penalties on 19 and 30 minutes, Aironi number eight Nick Williams sent to the sin bin for the second offence.
But the visitors, with their powerful pack to the fore, made light of their numerical disadvantage and got their first points through the boot of fly-half Naas Olivier after scrummaging their way to a penalty to make it 13-3 at half-time.
An ankle tap denied replacement Dragons wing Matthew Pewtner early in the second half and Olivier moved the Italians to within a score with a 47th-minute penalty.
The hosts were unable to shake off their visitors and were being hindered by a high error count in slippery conditions.
Aironi were growing in confidence with Williams and Samoan wing Sinoti Sinoti making plenty of hard yards and they made it 13-11 when lock Josh Furno barged over on 62 minutes.
But the home support were breathing easier when number eight Andrew Coombs stretched over in the corner with Tovey crucially booting the extras for a 20-11 lead with 13 minutes left.
Olivier reduced the deficit with a booming kick on 71 minutes but replacement full-back Matthew Jones thumped over a penalty to seal the home win with five left.
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