The real-time risk-averse dispatch problem of an integrated electricity and natural gas system (IEGS) is studied in this paper. It is formulated as a real-time conditional value-at-risk (CVaR)-based risk-averse dispatch model in the Markov decision process framework. Because of its stochasticity, nonconvexity and nonlinearity, the model is difficult to analyze by traditional algorithms in an acceptable time. To address this non-deterministic polynomial-hard problem, a CVaR-based lookup-table approximate dynamic programming (CVaR-ADP) algorithm is proposed, and the risk-averse dispatch problem is decoupled into a series of tractable subproblems. The line pack is used as the state variable to describe the impact of one period's decision on the future. This facilitates the reduction of load shedding and wind power curtailment. Through the proposed method, real-time decisions can be made according to the current information, while the value functions can be used to overview the whole optimization horizon to balance the current cost and future risk loss. Numerical simulations indicate that the proposed method can effectively measure and control the risk costs in extreme scenarios. Moreover, the decisions can be made within 10 s, which meets the requirement of the real-time dispatch of an IEGS.