We construct a constant curvature analogue on the two-dimensional sphere {\mathbf S}^2 and the hyperbolic space {\mathbf H}^2 of the integrable H\'enon-Heiles Hamiltonian \mathcal{H} given by \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ \Omega \left( q_{1}^{2}+ 4 q_{2}^{2}\right) +\alpha \left( q_{1}^{2}q_{2}+2 q_{2}^{3}\right) , where \Omega and \alpha are real constants. The curved integrable Hamiltonian \mathcal{H}_\kappa so obtained depends on a parameter \kappa which is just the curvature of the underlying space, and is such that the Euclidean H\'enon-Heiles system \mathcal{H} is smoothly obtained in the zero-curvature limit \kappa\to 0. On the other hand, the Hamiltonian \mathcal{H}_\kappa that we propose can be regarded as an integrable perturbation of a known curved integrable 1:2 anisotropic oscillator. We stress that in order to obtain the curved H\'enon-Heiles Hamiltonian \mathcal{H}_\kappa, the preservation of the full integrability structure of the flat Hamiltonian \mathcal{H} under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani-Dorizzi-Grammaticos (RDG) series \cal{V}_{n} of integrable polynomial potentials, in which the flat H\'enon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved RDG potentials \cal{V}_{\kappa, n} on {\mathbf S}^2 and {\mathbf H}^2 will be also explicitly presented.