If by scientific method one means observation and experiments then I would say "yes" in Japan. Harder to say for certain about China.

Sugita Genpaku (1733-1817) and others discovered that Western pictures of the human anatomy was different from Chinese diagrams. From there they conducted autopsies and checked for themselves in 1771. Satisfied that the Western pictures were more accurate they decided to publish in Japanese a translation of a Dutch textbook on the anatomy in 1774.

From there "Dutch studies" began to spread in Japan. By 1849 Sakuma Shozan (1811-1864) could replicate and successfully conduct experiments with electricity and the telegraph. [Samuel Morse invented the modern telegraph ca1835.] All this happened while Japan still maintained the so-called "Sakoku" (closed country) policy.

So for Japan, the scientific method (if understood to mean observation and experiment) was practiced by scholars in the late 1700s. I think they could have understood it in 1714 if they had encountered it.

edited to include Samuel Morse.

The scientific method is not a static thing that was ever "discovered". The gradual evolution of what is today known as the scientific method is more a philosophical commentary on science rather than something of primary concern to scientists themselves. This isn't to say that philosophy of science is irrelevant -- philosophical ideas definitely influence science, as Mill's and Popper's ideas certainly have. And some science, like Einstein's relativity, is hard to distinguish from philosophy. But its an ongoing dialog rather than a formalized practice that unifies scientific inquiry. And science in practice is always uglier and more complicated than the clean formalism that is described by various methods.

If by scientific method you mean a general process of testing ideas against evidence, and when push comes to shove trusting evidence over theory, this is a surprisingly modern idea. A historical vignette might help. When Newton's ideas of gravitation were published, a while after the idea of science was modestly developed and accepted (nearly two hundred years after Copernicus), they ran into trouble from Descartes and his school of science. Descartes claimed that Newton's ideas were "unscientific" because they posited these ridiculous "forces acting at a distance". Descartes was of the school that scientific truths could be reasoned and had to follow from known philosophical truths. (Which is probably what led him to unification of algebra and geometry among other accomplishments, so he wasn't wrong.) Newton responded that he couldn't explain his forces, but his theory worked because it agreed with evidence. (Which was also how Copernicus's ideas were originally introduced to a skeptical, religious public.) Over time, Newton won. Nobody still can really explain forces in terms of direct, tangible experience, but they are no longer a controversial scientific notion because evidence trumps theory. But that debate was still very much alive in the 18th century, and has never really died.

In my opinion, the scientific method is better thought of as a system of cultural norms that have emerged around the empirical pursuit of knowledge. These norms are tested and updated constantly in order to find better ways of describing the world. But this centuries-long process is a recent development and one not found in other eras.

Appeals to supernatural or philosophical philosophical explanations are the historical norm. Early Greek thinkers developed some proto-science, but it was overtaken by Aristotle and his followers that did not rigorously test their ideas against experiment. I'm not aware of strong empirical traditions developing in other cultures, but I'm happy to be enlightened.