Of course, the exact definition of the scientific method is open to discussion (and the scientific method may be very different in different sciences, compare chemistry, astronomy, geology
and mathematics, for example).
The earliest account that I know of an experimental discovery is physics is credited to Pythagoras. According to the legend, he once visited a blacksmith workshop and noticed that
similar metal rods of various length produce sounds of various pitch when struck with a hummer.
Apparently he experimented with various rods and strings, and derived the correct law relating the musical intervals with ratios of length of the strings that produce them.
Apparently this discovery impressed him so much that he concluded that "numbers rule the world", which is essentially the main paradigm of exact scienses. In modern formulation: the laws of nature are expressed in mathematical form.
This is what the legend says about the origin of ("Western European") exact science.
This happened some time in VI cent. BC.
Somewhat earlier in the same century mathematical method was born: Thales of Miletus proved the first theorems.
Of course, all this information comes from much later secondary sources. So these are property called "legends". But this is the best information we apparently possess.
According to Lucio Russo, with whom I agree, this led to the full development of the scientific method which reached its culmination in II BC and was still practiced in II AD. But then there was an interruption, and all exact science, together with the method were lost almost for a
It had to be reborn, rediscovered, and this happened in 16-th century,
when Francis Bacon
explicitly described the newly discovered scientific method. Fortunately, some of the Hellenistic science was preserved outside of Europe.
EDIT. I was asked to add sources.
For the story with Pythagoras, see "Pythagorean hummers" on Wikipedia, it lists original sources.
On Thales, his story is told in almost every history of Greek mathematics, I used van der Waerden, Science awakening.(English tranls. Ocford Univ press 1961). He also has a book on Pythagoreans, but I do not have it besides me.
Lucio Russo's book is called Forgotten revolution (engl. transl. Springer 2004).
There is a nice long review of this book in the Notices Amer. Math. Soc., which I believe is freely available. There is about a dozen other reviews online.
Francis Bacon's main work on scientific method is called New Organon (Wikipedia article "Francis Bacon" has a link to a free online version, and actually to cmplete works of Bacon).