Higher-order connectivity patterns such as small induced sub-graphs called graphlets (network motifs) are vital to understand the important components (modules/functional units) governing the configuration and behavior of complex networks. Existing work in higher-order clustering has focused on simple homogeneous graphs with a single node/edge type. However, heterogeneous graphs consisting of nodes and edges of different types are seemingly ubiquitous in the real-world. In this work, we introduce the notion of typed-graphlet that explicitly captures the rich (typed) connectivity patterns in heterogeneous networks. Using typed-graphlets as a basis, we develop a general principled framework for higher-order clustering in heterogeneous networks. The framework provides mathematical guarantees on the optimality of the higher-order clustering obtained. The experiments demonstrate the effectiveness of the framework quantitatively for three important applications including (i) clustering, (ii) link prediction, and (iii) graph compression. In particular, the approach achieves a mean improvement of 43x over all methods and graphs for clustering while achieving a 18.7% and 20.8% improvement for link prediction and graph compression, respectively.