During a conversation, there can come certain moments where its outcome hangs in the balance. In these pivotal moments, how one responds can put the conversation on substantially different trajectories leading to significantly different outcomes. Systems that can detect when such moments arise could assist conversationalists in domains with highly consequential outcomes, such as mental health crisis counseling. In this work, we introduce an unsupervised computational method for detecting such pivotal moments as they happen, in an online fashion. Our approach relies on the intuition that a moment is pivotal if our expectation of the outcome varies widely depending on what might be said next. By applying our method to crisis counseling conversations, we first validate it by showing that it aligns with human perception -- counselors take significantly longer to respond during moments detected by our method -- and with the eventual conversational trajectory -- which is more likely to change course at these times. We then use our framework to explore the relation of the counselor's response during pivotal moments with the eventual outcome of the session.