For a given weight function \(\omega: [0,\infty)\to [0,\infty)\) and an open set \(\Omega\subseteq \mathbb{R}^n\), the authors denote by \({\mathcal E}_{(\omega)}(\Omega)\) the non-quasianalytic class of Beurling type on \(\Omega\). They investigate conditions guaranteeing the surjectivity of the convolution operator \(T_\mu:{\mathcal E}_{(\omega)}(\Omega_1)\to {\mathcal E}_{(\omega)}(\Omega_2)\) for the given \(\mu\in{\mathcal E}_{(\omega)}'(\mathbb{R}^n)\). Analogous results are obtained also for ultradistributions of Roumieu type \({\mathcal D}_{(\omega)}'(\Omega)\).