We show the improvement to cosmological constraints from galaxy cluster surveys with the addition of cosmic microwave background (CMB)-cluster lensing data. We explore the cosmological implications of adding mass information from the 3.1σ detection of gravitational lensing of the CMB by galaxy clusters to the Sunyaev-Zel'dovich (SZ) selected galaxy cluster sample from the 2500 deg2 SPT-SZ survey and targeted optical and X-ray follow-up data. In the ΛCDM model, the combination of the cluster sample with the Planck power spectrum measurements prefers σ8ωm/0.30.5=0.831±0.020 . Adding the cluster data reduces the uncertainty on this quantity by a factor of 1.4, which is unchanged whether the 3.1σ CMB-cluster lensing measurement is included or not. We then forecast the impact of CMB-cluster lensing measurements with future cluster catalogs. Adding CMB-cluster lensing measurements to the SZ cluster catalog of the ongoing SPT-3G survey is expected to improve the expected constraint on the dark energy equation of state w by a factor of 1.3 to σ(w) = 0.19. We find the largest improvements from CMB-cluster lensing measurements to be for σ 8, where adding CMB-cluster lensing data to the cluster number counts reduces the expected uncertainty on σ 8 by respective factors of 2.4 and 3.6 for SPT-3G and CMB-S4.