Heterogeneous networks are seemingly ubiquitous in the real world. Yet, most graph mining methods such as clustering have mostly focused on homogeneous graphs by ignoring semantic information in real-world systems. Moreover, most methods are based on first-order connectivity patterns (edges) despite that higher-order connectivity patterns are known to be important in understanding the structure and organization of such networks. In this work, we propose a framework for higher-order spectral clustering in heterogeneous networks through the notions of typed graphlets and typed-graphlet conductance. The proposed method builds clusters that preserve the connectivity of higher-order structures built up from typed graphlets. The approach generalizes previous work on higher-order spectral clustering. We theoretically prove a number of important results including a Cheeger-like inequality for typed-graphlet conductance that shows near-optimal bounds for the method. The theoretical results greatly simplify previous work while providing a unifying theoretical framework for analyzing higher-order spectral methods. Empirically, we demonstrate the effectiveness of the framework quantitatively for three important applications including clustering, compression, and link prediction.