Ross Wardle and Wayne Evans scored in the space of four minutes as Dragons were forced to overturn a 9-3 half-time deficit at Stadio Maurizio Quaggia.
Steffan Jones levelled the scores, with prop Nathan Buck in the sin-bin, before the two tries and five points from replacement Kris Burton, but the Dragons could not grab a bonus even though Mogliano finished the game with 13 men.
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