Supernovae probably are the dominant energy source for driving turbulence in spiral galaxies. Until recently, their effect on magnetic field amplification in the interstellar medium was, however, poorly understood. Analytical models based on uncorrelated ensembles of SNe predicted that the created field would be expelled from the disk before it is significantly amplified. By means of self-consistent simulations of supernova-driven turbulence, we demonstrate that this is not the case. Accounting for vertical stratification and galactic differential rotation, we find an exponential amplification of the mean field on timescales of 100Myr. The direct numerical verification of such a fast dynamo is highly beneficial in explaining the observed strong magnetic fields in high red-shift galaxies. We, furthermore, highlight the importance of rotation in the generation of helicity by showing that a similar mechanism based on Cartesian shear alone does not lead to a sustained amplification of the mean magnetic field. This finding impressively confirms the classical picture of a dynamo based on cyclonic turbulence.