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Data from several large elliptical and disk galaxies now show that globular clusters more massive than ~10%5% M_sun_ follow a power-law number distribution by mass, N ~ M^-1.7^, which is virtually independent of environment. Within observational uncertainty, this relation is identical to the shape of the mass distributions of giant molecular clouds (GMCs) in large spiral galaxies, the cloud cores embedded in GMCs, and giant H II regions in large spiral galaxies. We interpret this within a model whereby globular clusters formed out of dense cores within supergiant molecular clouds (SGMCs) that were present in the early protogalactic epoch. We construct a theory of pressure confined, self- gravitating, isothermal, magnetized molecular clouds and cores, based on the virial theorem and the observed mass spectra, and derive the characteristic physical properties of these parent SGMCs. These turn out to be of the right mass and density range to resemble the Searle-Zinn primordial fragments from which larger galaxies may have assembled. We suggest that the protocluster clouds were supported against gravitational collapse primarily by a combination of magnetic field pressure and Alfvenic turbulence, as is observed to be the case for contemporary molecular clouds. This approach removes the need for arbitrary external heat sources (such as long-lasting AGNs or Population III stars) to keep the clouds stable for long enough times to build up to globular-sized masses and more easily permits the global properties of the emergent clusters to be similar from one galaxy to another. By calculating lifetimes through a standard cloud growth model, we estimate that the principal epoch of globular cluster formation should have begun no earlier than a redshift of z ~ 6.