Observations of non-thermal emission from several supernova remnants suggest that magnetic fields close to the blast wave are much stronger than would be naively expected from simple shock compression of the field permeating the interstellar medium (ISM). We present a simple model which is capable of achieving sufficient magnetic field amplification to explain the observations. We propose that the cosmic ray pressure gradient acting on the inhomogeneous ISM upstream of the supernova blast wave induces strong turbulence upstream of the supernova blast wave. The turbulence is generated through the differential acceleration of the upstream ISM which occurs as a result of density inhomogeneities in the ISM. This turbulence then amplifies the pre-existing magnetic field. Numerical simulations are presented which demonstrate that amplification factors of 20 or more are easily achievable by this mechanism when reasonable parameters for the ISM and supernova blast wave are assumed. The length scale over which this amplification occurs is that of the diffusion length of the highest energy non-thermal particles. © 2012 The Authors Monthly Notices of the Royal Astronomical Society © 2012 RAS.