If q < p, then the number of Sylow p-groups is = 1 (mod p), and divides q. Of course then, the number is 1, which means the Sylow p-group is normal in G.

If p < q, then the number of Sylow q-groups = 1 (mod q) and divides p^2. Therefore, since p<q, either there is one Sylow q-group, or p^2 many. If there is one, we are done. If there are p^2, then there are p^2 * (q-1) elements of order q in G, which means there are |G| - p^2 * (q-1) = p^2 elements in G that are not of order q. These elements must lie in a Sylow p-group, and since there are exactly p^2, there can only be one Sylow p-group, and thus it is normal.

Steve