To phase out fossil fuels, energy systems must shift to renewable electricity as the main source of primary energy. In this paper, we analyze how electrification can support the integration of fluctuating renewables, like wind and PV, and mitigate the need for storage and thermal backup plants. Using a cost-minimizing model for system planning, we find substantial benefits of electricity demand in heating, transport, and industry adapting to supply. In Germany, flexible demand halves the residual peak load and the residual demand and reduces excess generation by 80%. Flexible operation of electrolyzers has the most significant impact accounting for 42% of the reduction in residual peak load and 59% in residual demand. District heating networks and BEVs also provide substantial flexibility, while the contribution of space and process heating is negligible. The results are robust to restrictions on the expansion of the transmission grid.