We investigate the irreversible growth of (2+1)-dimensional magnetic thin films under the influence of a transverse temperature gradient, which is maintained by thermal baths across a direction perpendicular to the direction of growth. Therefore, different longitudinal layers grow at different temperatures between T₁ and T₂, where T₁ < T _c^hom < T₂ and T_c^hom = 0.69(1) is the critical temperature of films grown in homogeneous thermal baths. We find a far-from-equilibrium continuous order-disorder phase transition driven by the thermal bath gradient. We characterize this gradient-induced critical behavior by means of standard finite-size scaling procedures, which lead to the critical temperature T<subc=0.84(2) and a new universality class consistent with the set of critical exponents ν=3/2, γ=5/2, and β=1/4. In order to gain further insight into the effects of the temperature gradient, we also develop a bond model that captures the magnetic films growth dynamics. Our findings show that the interplay of geometry and thermal bath asymmetries leads to growth bond flux asymmetries and the onset of transverse ordering effects that explain qualitatively the shift observed in the critical temperature. The relevance of these mechanisms is further confirmed by a finite-size scaling analysis of the interface width, which shows that the growing sites of the system define a self-affine interface.Instituto de Física de Líquidos y Sistemas Biológicos