Although real-world complex systems typically interact through sparse and heterogeneous networks, analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of nonlinear models of complex systems on sparse directed networks with a random structure. By generalizing dynamical mean-field theory to sparse systems, we derive an exact equation for the path probability describing the effective dynamics of a single degree of freedom. Our general solution applies to key models in the study of neural networks, ecosystems, epidemic spreading, and synchronization. Using the population dynamics algorithm, we solve the path-probability equation to determine the phase diagram of a seminal neural network model in the sparse regime, showing that this model undergoes a transition from a fixed-point phase to chaos as a function of the network topology.