The spaces that you are looking for are precisely the minimal Hausdorff spaces. i.e. A Hausdorff space $X$ is a minimal Hausdorff if every injective continuous map from $X$ to a Hausdorff space is an embedding. See the Book General Topology by Stephen Willard problems 17M for more information about these spaces. The Minimal Hausdorff spaces are precisely the Hausdorff spaces where every open filter with a unique accumulation point converges.
We say that a topological space is semiregular if the regular open sets form a basis for the topology. A Hausdorff space is said to be $H$-closed if it is closed in every Hausdorff extension. Every minimal Hausdorff space is $H$-closed, and a Hausdorff space is a minimal Hausdorff space if and only if it is semiregular and $H$-closed.
While there are minimal Hausdorff spaces which are not compact, the notions of minimal Hausdorff and compactness coincide for regular spaces.