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We devise a machine learning technique to solve the general problem of inferring network links with time-delays purely from time-series data of the network nodal states. This task has applications in fields ranging from applied physics and engineering to neuroscience and biology. To achieve this, we train a type of machine learning system known as reservoir computing to mimic the dynamics of the unknown network and relate the parameters of the reservoir system to the unknown network structure. Our technique, by its nature, is non-invasive, but is motivated by the widely-used invasive network inference method whereby the responses to active perturbations applied to the network are observed and employed to infer network links (e.g., knocking down genes to infer gene regulatory networks). We test this technique on experimental and simulated data from delay-coupled opto-electronic oscillator networks. We show that the technique often yields very good results particularly if the system does not exhibit synchrony of the nodal dynamics. We also find that the presence of dynamical noise can strikingly enhance the accuracy and ability of our technique.