The 37-year-old, who put out Alex de Mnaur earlier in the event, is still playing within himself as he bids to build his fitness for the French Open next month.
He cruised through the first set but lost a scrappy second in a tie-break before eventually sealing a 6-1 6-7 (5-7) 6-3 victory in a little more than three hours.
Five-time winner Nadal, playing at his home tournament for probably the last time, said on Sky Sports: "I found a way to be through. In the third I was able to be more unpredictable and I think that changed the match.
"I'm enjoying the fact that I'm playing at home because it means everything to me and I'm trying my best to keep dreaming - and tomorrow is another day."
Play was briefly halted in the first set while a spectator required medical treatment, with Cachin rushing to the players' fridge to send bottles of water into the stand.
When they resumed Nadal promptly secured a third break of serve to wrap up the set.
But Cachin began swinging at the start of the second and broke the Nadal serve twice to lead 4-1 and silence the sell-out Spanish crowd.
The 22-time grand slam champion upped the intensity and retrieved one break before saving two set points in a marathon 10th game to level the set at 5-5 only to lose the tie-break.
However, Nadal secured a break at the start of the third and ultimately claimed victory to set up a fourth-round meeting with 22-year-old Czech Jiri Lehecka.
World No.4 Daniil Medvedev dropped the first set against American Sebastian Korda but hit back to win 5-7 7-6 (7-4) 6-3.
He will face Alexander Bublik of Kazakhstan, who beat up-and-coming American Ben Shelton in three sets, in the last 16.
Briton Cameron Norrie exited in straight sets after a 2-6 4-6 defeat to Norway's Casper Ruud.
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