Italy required a 17-point margin to sneak ahead of Wales into fourth and after Rigoni completed her double before the break and Michela Sillari added a penalty for a 13-0 lead, they looked on course for an unlikely coup.
Sioned Harries' scores either side of half-time took Wales to within a point, but Sillari's second penalty proved a match-winner.
It was the Italian's third consecutive RBS 6 Nations victory over Wales and Wales' second loss on the bounce since defeating Women's Six Nations Champions France at The Gnoll.
Italy came out firing and looked to have the bit between their teeth as they set out to try and open up the large gap needed to overhaul Wales for a fourth-placed finish.
Rigoni capped an exceptional first ten minutes by finishing off a well-constructed Italian back's move to go over for Italy, but Sillari could not add the extras.
Wales began to work themselves into the game, Rachel Taylor was exceptional for the hosts - making the hard yards when needed and putting the Italians on the back foot.
Just as Edwards' team started to look threatening, Rigoni rose to the occasion again. She perfectly read the Welsh play to intercept and gallop 90 metres to score Italy's second and stretch the lead to ten points.
Sillari put thoughts of two missed conversions to bed as she knocked over a penalty from directly in front of the posts to leave Wales with a mountain to climb at 13-0 down.
Thankfully for Wales, their ferocious pack took the game by the scruff of the neck and as Robyn Wilkins directed her side to the corner, Harries was the beneficiary of a perfectly formed maul which rifled irresistibly through the Italian pack and over to reduce the deficit to 13-5 at half-time, but Wilkins could not convert.
Wales' forwards started the second-half as they ended the first, pummelling the Italian line with a bludgeoning driving maul and it was Harries again who went over at the back with Wilkins adding the extras to bring the hosts within a point.
Sillari struck again with 25 minutes to go to establish a four-point lead and ease the pressure.
But it was Wales who seized the initiative yet again with Taylor thundering forward in the contact and Elinor Snowsil prominent and Italy looked to be struggling to hang on.
Yet the Italians defended superbly to keep the Welsh at bay, Sara Barattin using every shred of her experience and Rigoni kicking deftly, forcing the hosts to attack fruitlessly from deep.
