Context. In Italian, you don't have to use a subject pronoun. This sentence could also mean I don't want you or her to open the window (tu apra, lei apra), or even 'I don't want to open the window' (io apra). All should be judged correct.
I tried 'I don't want the window open', which was judged wrong. What's the right way to say this? Non voglio la finestra aperta?
In the translation that DL gives "I do not want him to open the window.", "he" is the subject of the verb "aprire - apra" not the window. And as Anthony says, the subject could just as easily be "him" or "her".
Yes, this is puzzling. The subject can certainly come after the verb in Italian (see, for example, http://www.scuolitalia.com/1/gram-syntax-subjectafterverb.htm), but so can the direct object. However, unless the window is one of those animated objects in a Disney movie, it likely can't open itself, but is more likely to be the object being opened by an admittedly non-explicit and therefore ambiguous subject (io? tu? lui? lei?). I therefore think that 'la finestra' is almost certainly the direct object of apra.
Thank you for detailed analysis and it make sense. 'I don't want that window open' was the incorrect English translation I had. For this to be correct answer, I later thought, then Italian sentence should have been 'non voglio che si apra la finestra'. But now I wonder if using 'apra' in reflexive construction is correct?