In this paper we follow the evolution of an unstable magnetized cloud core modeled with the density distribution of a singular isothermal sphere and threaded by a uniform magnetic field. We include neutral-ion slip, and we solve the equations by an expansion about the known self-similar problem without magnetism. We find that the magnetic field does not significantly modify the standard rate of mass infall because of two offsetting effects: the Lorentz force that impedes gravitational collapse, and the increased characteristic speed that causes the initiation of infall to travel outward faster (as a fast magnetohydrodynamic wave rather than an acoustic wave). Strong magnetic pinching forces deflect infalling gas toward the equatorial plane to form a flattened disequilibrium structure (''pseudodisk'') around the central protostar. The perturbative approach allows us to calculate analytically the dependence of the radius of the pseudodisk at small times on the physical parameters of the problem when a dimensionless coefficient of order unity is supplied by a separate numerical calculation for the nonlinear flow in the inner region (Paper II).