We consider the the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$ in dimensions $d \ge 3$. For varying intensity $u > 0$, the connectivity properties of $\mathcal V^u$ undergo a percolation phase transition across a critical parameter $u_* \in (0,\infty)$. In this article, we prove that this phase transition is sharp in the supercritical phase $u < u_*$. This follows from a certain strong local uniqueness property (SLU) introduced in the present work, which we prove $\mathcal{V}^u$ satisfies. In itself, this property furnishes the missing ingredient needed to deduce a number of desirable quenched results characterizing the large-scale geometry of the infinite cluster. Moreover, SLU entails a sought-after local and monotone criterion amenable to renormalization arguments below $u_*$.