We study how tunneling through a potential barrier V(x) can be enhanced by an additional harmonically oscillating electric field E(t)=E0cos(ωt) or a similar oscillating force. To this end, we transform into the Kramers-Henneberger frame and calculate the coupled Floquet channels numerically. We find distinct signatures of resonances when the incident energy E equals the driving frequency ω=E, which clearly shows the breakdown of the time-averaged potential approximation in this regime. As a simple model for experimental applications (e.g., in solid-state physics), we study the rectangular potential, which can also be benchmarked with respect to analytical results. Finally, we consider the truncated Coulomb potential relevant for nuclear fusion.