We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $\mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations: a flat fibration $\mathbb{A}^m$ → $\mathbb{A}^n$ with all fibers isomorphic to $\mathbb{A}^2$ is the trivial $\mathbb{A}^2$-bundle.