Clusters of galaxies are expected to gravitationally lens the cosmic microwave background (CMB) and thereby generate a distinct signal in the CMB on arcminute scales. Measurements of this effect can be used to constrain the masses of galaxy clusters with CMB data alone. Here we present a measurement of lensing of the CMB by galaxy clusters using data from the South Pole Telescope (SPT). We develop a maximum likelihood approach to extract the CMB cluster lensing signal and validate the method on mock data. We quantify the effects on our analysis of several potential sources of systematic error and find that they generally act to reduce the best-fit cluster mass. It is estimated that this bias to lower cluster mass is roughly 0.85\sigma in units of the statistical error bar, although this estimate should be viewed as an upper limit. We apply our maximum likelihood technique to 513 clusters selected via their SZ signatures in SPT data, and rule out the null hypothesis of no lensing at 3.1\sigma. The lensing-derived mass estimate for the full cluster sample is consistent with that inferred from the SZ flux: M_{200,\mathrm{lens}} = 0.83_{-0.37}^{+0.38}\, M_{200,\mathrm{SZ}} (68% C.L., statistical error only). Show Partial Abstract