The Ewens sampling formula is a family of measures on permutations, that arises in population genetics, Bayesian statistics and many other applications. This family is indexed by a parameter $\theta > 0$; the usual uniform measure is included as the special case $\theta = 1$. Under the Ewens sampling formula with parameter $\theta$, the process of cycle counts $(C_1(n), C_2(n), \ldots, C_n(n), 0, 0, \ldots)$ converges to a Poisson process $(Z_1, Z_2, \ldots)$ with independent coordinates and $\mathbb{E}Z_j = \theta/j$. Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of $(C_1(n), \ldots, C_b(n))$ and $(Z_1, \ldots, Z_b)$. This Poisson approximation can be used to give simple proofs of limit theorems with bounds for a wide variety of functionals of such random permutations.