We describe a finite-dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite-dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.