"do cachorro" can be possessive, too, right? And just the more plausible context tells us that it is "saves the cat from the dog" and not "saves the dog's cat"? Is there another way to say one of these so that it can be less ambiguous when needed? The dog as cat-owner part may be ridiculous, but what about, e.g., "She saves the woman's cat" vs. "She saves the cat from the woman"? "Ela salva o gato da mulher."
Well, most of the time, it depends on the context, and I do agree English is clearer in this regard when compared to Portuguese.
For "She saves the cat from the woman'", you could say for example "Ele salva o gato das garras da mulher".