Core-collapse supernovae produce fast shocks which expand into the dense circumstellar medium (CSM) of the stellar progenitor. Cosmic rays (CRs) accelerated at these shocks can induce the growth of electromagnetic fluctuations in the pre-shock medium. Using a self-similar description for the shock evolution, we calculate the growth time-scales of CR driven instabilities for SNe in general, and SN 1993J in particular. We find that extended SN shocks can trigger fast intra-day instabilities, strong magnetic field amplification, and CR acceleration. In particular, the non-resonant streaming instability can contribute to about 50 per cent of the magnetic field intensity deduced from radio data. This results in the acceleration of CR particles to energies of 1-10 PeV within a few days after the shock breakout.