Importance of having a "connection" while applying to PhD

I am applying for a PhD to top schools in engineering. Whenever I talk to my peers I hear the word 'connection' very often. I started thinking that it is almost impossible to be admitted to these schools without knowing a professor ! (I am an international applicant)

How important is having a connection when applying? What If I don't know one and I believe that my profile qualifies me for admission? Should I apply or just not bother?