The problem of multihypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can reduce the delay by intelligently choosing the control for observation collection in each time slot. A policy for this problem is derived that satisfies given constraints on the error probability, and it is shown that this policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Numerical results are provided that illustrate an application of the policy to medical diagnostic inference. © 2021 Taylor & Francis Group, LLC.