We present Optimal Completion Distillation (OCD), a training procedure for optimizing sequence to sequence models based on edit distance. OCD is efficient, has no hyper-parameters of its own, and does not require pretraining or joint optimization with conditional log-likelihood. Given a partial sequence generated by the model, we first identify the set of optimal suffixes that minimize the total edit distance, using an efficient dynamic programming algorithm. Then, for each position of the generated sequence, we define a target distribution that puts an equal probability on the first token of each optimal suffix. OCD achieves the state-of-theart performance on end-to-end speech recognition, on both Wall Street Journal and Librispeech datasets, achieving 9.3% and 4.5% word error rates, respectively.