We study the Swendsen-Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a non-local Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work we derive strong enough bounds on the mixing time, proving that the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures exhibits a sharp transition from "unmixed" to "well-mixed," which is called the cutoff phenomenon. In particular, we establish that at high enough temperatures the Swendsen-Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^d$ has cutoff at time $\frac{d}{2} \left( -\log (1-γ) \right)^{-1} \log n$, where $γ(β)$ is the spectral gap of the infinite-volume dynamics.