We use measurements from the South Pole Telescope (SPT) Sunyaev-Zel'dovich (SZ) cluster survey in combination with X-ray measurements to constrain cosmological parameters. We present a statistical method that fits for the scaling relations of the SZ and X-ray cluster observables with mass while jointly fitting for cosmology. The method is generalizable to multiple cluster observables, and self-consistently accounts for the effects of the cluster selection and uncertainties in cluster mass calibration on the derived cosmological constraints. We apply this method to a data set consisting of an SZ-selected catalog of 18 galaxy clusters at z > 0.3 from the first 178 deg{sup 2} of the 2500 deg{sup 2} SPT-SZ survey, with 14 clusters having X-ray observations from either Chandra or XMM-Newton. Assuming a spatially flat {Lambda}CDM cosmological model, we find the SPT cluster sample constrains {sigma}{sub 8}({Omega} {sub m}/0.25){sup 0.30} = 0.785 {+-} 0.037. In combination with measurements of the cosmic microwave background (CMB) power spectrum from the SPT and the seven-year Wilkinson Microwave Anisotropy Probe data, the SPT cluster sample constrains {sigma}{sub 8} = 0.795 {+-} 0.016 and {Omega} {sub m} = 0.255 {+-} 0.016, a factor of 1.5 improvement on each parameter over the CMB data alone. We consider several extensions beyond the {Lambda}CDM model by including the following as free parameters: the dark energy equation of state (w), the sum of the neutrino masses ({Sigma}m {sub {nu}}), the effective number of relativistic species (N {sub eff}), and a primordial non-Gaussianity (f {sub NL}). We find that adding the SPT cluster data significantly improves the constraints on w and {Sigma}m {sub {nu}} beyond those found when using measurements of the CMB, supernovae, baryon acoustic oscillations, and the Hubble constant. Considering each extension independently, we best constrain w = -0.973 {+-} 0.063 and the sum of neutrino masses {Sigma}m {sub {nu}} < 0.28 eV at 95% confidence, a factor of 1.25 and 1.4 improvement, respectively, over the constraints without clusters. Assuming a {Lambda}CDM model with a free N {sub eff} and {Sigma}m {sub {nu}}, we measure N {sub eff} = 3.91 {+-} 0.42 and constrain {Sigma}m {sub {nu}} < 0.63 eV at 95% confidence. We also use the SPT cluster sample to constrain f {sub NL} = -220 {+-} 317, consistent with zero primordial non-Gaussianity. Finally, we discuss the current systematic limitations due to the cluster mass calibration, and future improvements for the recently completed 2500 deg{sup 2} SPT-SZ survey. The survey has detected {approx}500 clusters with a median redshift of {approx}0.5 and a median mass of {approx}2.3 Multiplication-Sign 10{sup 14} M {sub Sun} h {sup -1} and, when combined with an improved cluster mass calibration and existing external cosmological data sets will significantly improve constraints on w.