Alaves, fourth with 32 points, hoped to put some distance between themselves and European champions Real Madrid, fifth on 30 points, who face Real Betis on Sunday.
However, Stuani's audacious early strike from Pedro Porro's cross sent Eusebio Sacristan's side ahead in an entertaining first half at Montilivi.
Baston levelled five minutes after the break, blasting home from close range after Jonathan Calleri's mis-hit shot looped into the air.
The forward came close to adding a second but could not finish at the back post under pressure from Ximo Navarro, while Choco Lozano had a chance to win it for Girona, provisionally eighth, but mistimed his header.
"The team never gives up, there's been a lot of games we've come back in. We got a point at a difficult ground," said Alaves striker Baston.
"We hope in the second half of the season we can keep adding up points, we'll keep working."
(Reporting by Rik Sharma, editing by Pritha Sarkar)
