The nexus of COVID-19 and climate change has so far brought attention to short-term greenhouse gas (GHG) emissions reductions, public health responses, and clean recovery stimulus packages. We take a more holistic approach, making five broad comparisons between the crises with five associated lessons for climate change mitigation policy. First, delay is costly. Second, policy design must overcome biases to human judgment. Third, inequality can be exacerbated without timely action. Fourth, global problems require multiple forms of international cooperation. Fifth, transparency of normative positions is needed to navigate value judgments at the science-policy interface. Learning from policy challenges during the COVID-19 crisis could enhance efforts to reduce GHG emissions and prepare humanity for future crises.