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In this work, we study the effect of a magnetic field on the growth of cosmological perturbations. We develop a mathematical consistent treatment in which a perfect fluid and a uniform magnetic field evolve together in a Bianchi I universe. We then study the energy density perturbations on this background with particular emphasis on the effect of the background magnetic field. We develop a full relativistic solution which refines previous analysis in the relativistic limit, recovers the known ones in the Newtonian treatment with adiabatic sound speed, and it adds anisotropic effects to the relativistic ones for perturbations with wavelength within the Hubble horizon. This represents a refined approach on the perturbation theory of an isotropic universe in GR, since most of the present studies deal with fully isotropic systems. [ABSTRACT FROM AUTHOR]