Quantifying the effectiveness of large-scale non-pharmaceutical interventions (NPIs) against COVID-19 is critical to adapting responses against future waves of the pandemic. Most studies of NPIs thus far have relied on epidemiological data. Here, we report the impact of NPIs on the evolution of SARS-CoV-2, taking the perspective of the virus. We examined how variations through time and space of SARS-CoV-2 genomic divergence rates, which reflect variations of the epidemic reproduction number Rt, can be explained by NPIs and combinations thereof. Based on the analysis of 5,198 SARS-CoV-2 genomes from 57 countries along with a detailed chronology of 9 non-pharmaceutical interventions during the early epidemic phase up to May 2020, we find that home containment (35% Rt reduction) and education lockdown (26%) had the strongest predicted effectiveness. To estimate the cumulative effect of NPIs, we modelled the probability of reducing Rt below 1, which is required to stop the epidemic, for various intervention combinations and initial Rt values. In these models, no intervention implemented alone was sufficient to stop the epidemic for Rt’s above 2 and all interventions combined were required for Rt’s above 3. Our approach can help inform decisions on the minimal set of NPIs required to control the epidemic depending on the current Rt value.