Genomic sequencing has significant potential to inform public health management for SARS-CoV-2. Here we report high-throughput genomics for SARS-CoV-2, sequencing 80% of cases in Victoria, Australia (population 6.24 million) between 6 January and 14 April 2020 (total 1,333 COVID-19 cases). We integrate epidemiological, genomic and phylodynamic data to identify clusters and impact of interventions. The global diversity of SARS-CoV-2 is represented, consistent with multiple importations. Seventy-six distinct genomic clusters were identified, including large clusters associated with social venues, healthcare and cruise ships. Sequencing sequential samples from 98 patients reveals minimal intra-patient SARS-CoV-2 genomic diversity. Phylodynamic modelling indicates a significant reduction in the effective viral reproductive number (Re) from 1.63 to 0.48 after implementing travel restrictions and physical distancing. Our data provide a concrete framework for the use of SARS-CoV-2 genomics in public health responses, including its use to rapidly identify SARS-CoV-2 transmission chains, increasingly important as social restrictions ease globally.