The problem of multi-hypothesis 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 probability of error is below a given constraint. The decision maker can control the delay by intelligently choosing the control for observation collection in each time slot. We derive a policy that satisfies the given constraint on the error probability. We also show that the policy is asymptotically optimal in the sense that it asymptotically achieves an information-theoretic lower bound on the expected delay. Sequential controlled sensing is a stochastic framework wherein a decision-maker collects observations from a set of controls by sequentially choosing a control and obtaining an observation associated with that control. This paradigm is encountered in information-gathering systems with multiple degrees of freedom that can be controlled adaptively to achieve a given statistical inference task. In traditional control systems, the control is responsible for governing the state of the system. On the other hand, in controlled sensing, the control governs the quality of observations.