The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century. In this paper we introduce a new notion of Wilf equivalence for integer partitions, and, using this notion, we prove that rook equivalence implies Wilf equivalence. We also prove that if we refine the notions of rook and Wilf equivalence in a natural way, then these two notions coincide. In [6] we prove that Wilf equivalence implies rook equivalence.