Deploying data- and computation-intensive applications such as large-scale AI into heterogeneous dispersed computing networks can significantly enhance application performance by mitigating bottlenecks caused by limited network resources, including bandwidth, storage, and computing power. However, current resource allocation methods in dispersed computing do not provide a comprehensive solution that considers arbitrary topology, elastic resource amount, reuse of computation results, and nonlinear congestion-dependent optimization objectives. In this paper, we propose LOAM, a low-latency joint communication, caching, and computation placement framework with a rigorous analytical foundation that incorporates the above aspects. We tackle the NP-hard aggregated cost minimization problem with two methods: an offline method with a 1/2 approximation and an online adaptive method with a bounded gap from the optimum. Through extensive simulation, the proposed framework outperforms multiple baselines in both synthesis and real-world network scenarios.