The Bianconeri were heading for back-to-back defeats after Inter recovered from a goal down, despite losing Matias Vecino to a red card, only for Gonzalo Higuain to add to Milan Skriniar's own goal late on.
A match full of talking points started with the defending champions edging in front 13 minutes in through Douglas Costa, who found the back of the net from a tight angle following a cross-field pass from Cuadrado.
VAR had to be used to ensure that an offside Blaise Matuidi did not get a touch in the build-up, and the system worked against Inter for a second time soon after to confirm that Vecino went in studs-up on Mario Mandzukic for a straight red.
Miralem Pjanic was fortunate to avoid being shown a second yellow for a challenge on Rafinha on the hour but, after Higuain fired wide with just Samir Handanovic to beat, the Nerazzurri incredibly found themselves in front.
Juve hit back, though, with Costa's shot being well saved and Paulo Dybala also asking serious questions of Handanovic once introduced from the bench.
Then came the late double, as Skriniar turned Cuadrado's cross into his own net before Higuain peeled away from his marker to convert Dybala's free kick from close range.
INTER MILAN (4-2-3-1): Handanovic; Cancelo, Skriniar, Miranda, D'Ambrosio (Karamoh 94'); Vecino, Brozovic; Candreva, Rafinha (Borja Valero 80'), Perisic; Icardi (Santon 85')
JUVENTUS (4-3-3): Buffon; Cuadrado, Barzagli, Rugani, Alex Sandro; Khedira (Dybala 61'), Pjanic (Bentancur 80'), Matuidi; Douglas Costa, Higuain, Mandzukic (Bernardeschi 66')