Germany has set an ambitious target of reaching net zero greenhouse gas emissions by 2045. We explore how integrated cross-sectoral planning can reduce costs compared to existing national plans. Our new linear optimization model PyPSA-DE simulates the electricity and hydrogen transmission networks, as well as supply, demand, and storage in all sectors of the energy system in Germany and its neighboring countries with high spatial and temporal resolution. While our new model shows strong electricity transmission grid development, total expansion is one third lower than in the national grid development plan, lowering costs by 92 billion €_2020 to 191 billion €_2020 and average grid tariffs by 7.5 €2020/MWh. These savings are mainly due to integrated planning and operation, a market design with regional prices, and a system-optimal usage of offshore wind. PyPSA-DE is open-source and can readily be adapted to study related issues around the energy transition.