We investigate the Minimal Theory of Massive Gravity (MTMG) in the light of different observational data sets which are in tension within the $\Lambda$CDM cosmology. In particular, we analyze MTMG model, for the first time, with the Planck-CMB data, and how these precise measurements affect the free parameters of the theory. The MTMG model can affect the CMB power spectrum at large angular scales and cause a suppression on the amplitude of the matter power spectrum. We find that on adding Planck-CMB data, the graviton has a small, positive, but non-zero mass at 68\% confidence level, and from this perspective, we show that the tension between redshift space distortions measurements and Planck-CMB data in the parametric space $S_8 - \Omega_m$ can be resolved within the MTMG scenario. Through a robust and accurate analysis, we find that the $H_0$ tension between the CMB and the local distance ladder measurements still remains but can be reduced to $\sim3.5\sigma$ within the MTMG theory. The MTMG is very well consistent with the CMB observations, and undoubtedly, it can serve as a viable candidate amongst other modified gravity theories.