Using three magnified Type Ia supernovae (SNe Ia) detected behind CLASH (Cluster Lensing and Supernovae with Hubble) clusters, we perform a first pilot study to see whether standardizable candles can be used to calibrate cluster mass maps created from strong lensing observations. Such calibrations will be crucial when next-generation Hubble Space Telescope cluster surveys (e.g. Frontier) provide magnification maps that will, in turn, form the basis for the exploration of the high-redshift Universe. We classify SNe using combined photometric and spectroscopic observations, finding two of the three to be clearly of Type Ia and the third probable. The SNe exhibit significant amplification, up to a factor of 1.7 at similar to 5 Sigma significance (SN-L2). We conducted this as a blind study to avoid fine-tuning of parameters, finding a mean amplification difference between SNe and the cluster lensing models of 0.09 +/- 0.09(stat) +/- 0.05(sys) mag. This impressive agreement suggests no tension between cluster mass models and high-redshift-standardized SNe Ia. However, the measured statistical dispersion of Sigma(mu) = 0.21 mag appeared large compared to the dispersion expected based on statistical uncertainties (0.14). Further work with the SN and cluster lensing models, post-unblinding, reduced the measured dispersion to Sigma(mu) = 0.12. An explicit choice should thus be made as to whether SNe are used unblinded to improve the model, or blinded to test the model. As the lensed SN samples grow larger, this technique will allow improved constraints on assumptions regarding e.g. the structure of the dark matter halo.