You're absolutely right that, if we define life generally as a self-replicating system implemented in chemistry, there is no a priori reason to assume that extraterrestrial life would use the same elements as we do. However, there are a couple good reasons that this might be the case:

Abundance - Broadly speaking, the prevalence of an element decreases with its atomic mass, which means that, while elemental availability will differ from planet to planet, lighter elements will always be more abundant. Thus, statistically speaking, it is likely that extraterrestrial life uses C,H,N,O just as we do.

Bonding - When you study the chemical bonding of elements in relation to their position in the periodic table, what you realize is the "first row," i.e. C,N,O, is actually an exception and the rest of the periodic table is the rule. What this means in practice is that these elements are capable of a much greater diversity in bonding than those below. It is this diversity that allows for the development of the complex biomolecular machinery necessary for life.

Edit: Here's a great essay by Nobel laureate Roald Hoffmann on the differences between Si and C and the implications for life: http://pubs.acs.org/cen/80th/print/silicon.html