Is it possible to defer graduate admission for one year? I got accepted into the Phd program at a Top 3 university in US. 

However, as I have an external scholarship (full funding for PhD study), I have to defer my admission to work at the scholarship sponsor research institute for one year. 

I wrote to the department explaining my situation, but was informed that deferral is not allowed and I have to reapply next year. 
Anybody had a similar experience as me? Is there a chance to appeal for my deferral request? 

As I have my own fundings, technically speaking I do not occupy a quota in current cohort.
You have no 'rights' in this matter.  You apply, they accept you or not, under their own terms.  So all you can do is ask again, maybe giving fuller details of your situation.