Changes in CO$_2$ emissions during the COVID-19 pandemic have been estimated from indicators on activities like transportation and electricity generation. Here, we instead use satellite observations together with bottom-up information to track the daily dynamics of CO$_2$ emissions during the pandemic. Unlike activity data, our observation-based analysis can be independently evaluated and can provide more detailed insights into spatially-explicit changes. Specifically, we use TROPOMI observations of NO$_2$ to deduce ten-day moving averages of NO$_x$ and CO$_2$ emissions over China, differentiating emissions by sector and province. Between January and April 2020, China's CO$_2$ emissions fell by 11.5% compared to the same period in 2019, but emissions have since rebounded to pre-pandemic levels owing to the fast economic recovery in provinces where industrial activity is concentrated.