Thanks, in the meanwhile I made a very similar reasoning, which I post below (partly for my own reference; I tend to lose my notes). Take any compactum $K \subset X$. Then $h(K)$ is compact by assumption. Thus any closed set $C\subset h(K)$ is also compact, being a closed subset of a compactum. Therefore $h^{-1}(C)$ is compact, and hence closed ($X$ is Hausdorff). Thus the pre-image of any closed set $C$ under $h|_K$ is closed, i.e. $h|_K$ is continuous. By compact generation, $h$ itself is continuous. Similarly, one shows that $h^{-1}$ is continuous, i.e. $h$ is a homeomorphism.