The problem also occurs in a plain Python installation.

The solution is simple.
Just copy the 2 DLLs from
C:\WinPython35\python-3.5.2\Lib\site-packages\pywin32_system32 to
C:\WinPython35\python-3.5.2\Lib\site-packages\win32.

For the plain Python the solution is the same (with the obvious change in paths).

The issue is reproducible with WinPython. I tested it with Anaconda as well and 
it works fine. It is probably related to setting some environment variables done 
by the myriad of WinPython scripts.

I use WinPython 3.5.2 and when running a simple py script:
import win32api
print('123')
everything runs in the Python shell, as it should.
However, if I try to run it in Eric6 it always returns an "unhandled import error".
I checked the sys.path and everything seems ok.