Stuani scored both goals to take his tally for the season to 15 in LaLiga, a mark only bettered by Barcelona duo Lionel Messi and Luis Suarez.
The Uruguay international opened the scoring on the half-hour mark when he poked home from Alex Granell's corner before finishing off 10-man Rayo in the 85th minute with a far-post header following a counter-attack.
That earned Girona a second successive away victory, after stunning Real Madrid 2-1 on their last trip to the capital a fortnight ago to end a 13-match winless run in all competitions.
Victory was no more than the Catalan outfit deserved, with Granell and Pere Pons Riera both hitting the woodwork, while Cristian Portu and Pedro Alcala also went close.
Rayo were not without their chances, with visiting goalkeeper Gorka Iraizoz doing well to deny Adrian Embarba and Raul De Tomas.
But the home side lost Abdoulaye Ba to a straight red card for a professional foul on the hour and could find no way back.