Most populated corners of the planet have been exposed to SARS-CoV-2, the coronavirus behind the COVID-19 pandemic. We examined the progression of COVID-19 in four island nations that fared well over the first three months of the pandemic: New Zealand, Australia, Iceland, and Taiwan. Using Bayesian phylodynamic methods, we estimated the effective reproduction number of COVID-19 in the four islands as 1-1.4 during early stages of the pandemic, and show that it declined below 1 as human movement was restricted. Our reconstruction of COVID-19's phylogenetic history indicated that this disease was introduced many times into each island, and that introductions slowed down markedly when the borders closed. Finally, we found that New Zealand clusters identified via standard health surveillance largely agreed with those defined by genomic data. Our findings can assist public health decisions in countries with circulating SARS-CoV-2, and support efforts to mitigate any second waves or future epidemics.