The hosts had taken the lead through Lionel Messi after just 19 minutes, but were rocked when Clement Lenglet was given his marching orders just after the half-hour mark.
Girona then took full advantage as Stuani stole in to bring his side back level.
A cross was sent in from the right by Aday Benitez more in hope than expectation, but with Gerard Pique on his heels, Stuani seized on it.
The Uruguayan nudged the ball past Pique and Busquets before lifting an effort over Ter Stegen and into the back of the net.
The strike was Stuani's fourth of the season so far as he continues his late-career purple patch.