With increasing attention to climate change, the penetration level of renewable energy sources (RES) in the electricity network is increasing. Due to the intermittency of RES, gas-fired power plants could play a significant role in backing up the RES in order to maintain the supply–demand balance. As a result, the interaction between gas and power networks are significantly increasing. On the other hand, due to the increase in peak demand (e.g., electrification of heat), network operators are willing to execute demand response programs (DRPs) to improve congestion management and reduce costs. In this context, modeling and optimal implementation of DRPs in proportion to the demand is one of the main issues for gas and power network operators. In this paper, an emergency demand response program (EDRP) is implemented locally to reduce the congestion of transmission lines and gas pipelines more efficiently. Additionally, the effects of optimal implementation of local emergency demand response program (LEDRP) in gas and power networks using linear and non-linear economic models (power, exponential and logarithmic) for EDRP in terms of cost and line congestion and risk of unserved demand are investigated. The most reliable demand response model is the approach that has the least difference between the estimated demand and the actual demand. Furthermore, the role of the LEDRP in the case of hydrogen injection instead of natural gas in the gas infrastructure is investigated. The optimal incentives for each bus or node are determined based on the power transfer distribution factor, gas transfer distribution factor, available electricity or gas transmission capability, and combination of unit commitment with the LEDRP in the integrated operation of these networks. According to the results, implementing the LEDRP in gas and power networks reduces the total operation cost up to 11% and could facilitate hydrogen injection to the network. The proposed hybrid model is implemented on a 24-bus IEEE electricity network and a 15-bus gas network to quantify the role and value of different LEDRP models.