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Network routing problems are common across many engineering applications. Computing optimal routing policies requires knowledge about network demand, i.e., the origin and destination (OD) of all requests in the network. However, privacy considerations make it challenging to share individual OD data that would be required for computing optimal policies. Privacy can be particularly challenging in standard network routing problems because sources and sinks can be easily identified from flow conservation constraints, making feasibility and privacy mutually exclusive. In this paper, we present a differentially private algorithm for network routing problems. The main ingredient is a reformulation of network routing which moves all user data-dependent parameters out of the constraint set and into the objective function. We then present an algorithm for solving this formulation based on a differentially private variant of stochastic gradient descent. In this algorithm, differential privacy is achieved by injecting noise, and one may wonder if this noise injection compromises solution quality. We prove that our algorithm is both differentially private and asymptotically optimal as the size of the training set goes to infinity. We corroborate the theoretical results with numerical experiments on a road traffic network which show that our algorithm provides differentially private and near-optimal solutions in practice.