Context. Molecular clouds are known to be turbulent and strongly affected by stellar feedback. Moreover, stellar feedback is believed to drive turbulence at large scales in galaxies. Aims. We study the role played by supernovae in molecular clouds and the influence of the magnetic field on this process. Methods. We performed three-dimensional numerical simulations of supernova explosions, in and near turbulent self-gravitating molecular clouds. In order to study the influence of the magnetic field, we performed both hydrodynamical and magnetohydrodynamical simulations. We also ran a series of simple uniform density medium simulations and developed a simple analytical model. Results. We find that the total amount of momentum that is delivered during supernova explosions typically varies by a factor of about 2, even when the gas density changes by 3 orders of magnitude. However, the amount of momentum delivered to the dense gas varies by almost a factor of 10 if the supernova explodes within or outside the molecular cloud. The magnetic field has little influence on the total amount of momentum injected by the supernova explosions but increases the momentum injected into the dense gas. Conclusions. Supernovae that explode inside molecular clouds remove a significant fraction of the cloud mass. Supernovae that explode outside have a limited influence on the cloud but nevertheless remove a substantial amount of gas at densities between 1 cm −3 and 100 cm −3 , that would be forming stars later. It is thus essential to know sufficiently well the correlation between supernovae and the surrounding dense material in order to know whether supernovae can regulate star formation effectively.