Toulouse went in at the break trailing 27-8 but second-half tries from Jules Plisson and Alexandre Flanquart put them back in it.
Yohan Montes then got another try in stoppage time with Jean-Marc Doussain's conversion sealing a draw.
Toulouse are currently in sixth position, just a point behind Stade.
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Elsewhere, third place Montpellier kept the pressure on the league leaders thanks to a 22-8 victory at Oyonnax.
Leaders Clermont beat second-place Toulon 22-16 on Friday night in the weekend's big game.
Castres are fourth and maintained their strong position with a 38-6 win over Brive.
Perpignan beat Biarritz 16-10 while Bordeaux edged out Bayonne 23-22.
Racing Metro had a good 26-13 away win over Grenoble in the late game.
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Bordeaux-Begles stun leaders Toulon