Space and water heating accounts for about 40% of final energy consumption in the European Union and thus plays a key role in reducing overall costs and greenhouse gas emissions. Many scenarios to reach net-zero emissions in buildings rely on electrification, but meeting the heat demand peaks in the winter can be challenging, particularly when wind and solar resources are low. This paper examines how to mitigate space heating demand peaks most cost-effectively in a top-down, sector-coupled model with carbon dioxide emissions constraint to be net-zero. It introduces the first model that co-optimises both supply and efficiency simultaneously including all European countries with hourly resolution. The competition between technologies to address these heating peaks, namely building retrofitting, thermal energy storage and individual hybrid heat pumps with backup gas boilers is examined. A novel thought experiment demonstrates that the level of building renovation is driven by the strong seasonal heat peaks, rather than the overall energy consumption. If all three instruments are applied, total costs are reduced by up to 17%. Building renovation enables the largest benefit with cost savings of up to 14% and allows individual gas boilers to be removed from the energy system without significant higher costs.