We study the quantum dynamics of a periodically driven biased two-state system which additionally is subject to either Ohmic or to frequency-dependent damping. Within the noninteracting-blip approximation for the stochastic forces, the solution is given in terms of a series which is defined by recursion relations. Avoiding rotating wave and Markovian approximation, we deduce the solution in closed form for the important cases of high- and low-frequency driving. For low-frequency driving, the dynamics is governed by a rate equation with a time-dependent rate and with a time-dependent adiabatic equilibrium state. In the high-frequency case, we find novel cooperative effects, such as resonances in the spectral distribution near fractional values of the static detuning energy.