Jonny Wilkinson had put the home side in the driving seat with six penalties in the first hour, but Julien Dupuy and Jerome Porical's three penalties between them along with Jules Plisson's drop goal kept Stade in touch.
And the capital club appeared to have pinched the match with eight minutes left - with Delon Armitage in the sin bin - as Hugo Bonneval scored a try, which Porical converted to put the visitors in the lead.
But Toulon were not finished, as the pressured to get back in front by forcing two penalties, both of which Wilkinson landed to give his side victory.
Clermont made sure not to lose ground on Toulon with a comfortable 18-11 win at Agen to remain second, five points adrift.
Toulouse and Montpellier also enjoyed easy wins to remain third and fourth respectively, Toulouse beating Racing Metro 32-13 and Montpellier beating Mont-de-Marsan 32-16.
In the day's other matches Castres edged an epic 38-36 against Perpignan thanks to Rory Kockott's 79th-minute penalty, Grenoble beat Biarritz 43-21 and Bayonne beat Bordeaux-Belges 22-11.