The integrated operation of the electricity and gas systems has attracted the attention of many researchers due to the ever-increasing interdependency between the two systems. In this paper, a novel framework for the real-time rolling dispatch of the integrated system is presented, targeting to attain the economically optimal and technically secure gas system real-time operation through the control of the available flexibility procured by various resources. A decoupled day-ahead scheduling is initially executed to determine unit commitment and gas linepack target decisions, which are then utilized as inputs to the proposed integrated real-time dispatch model. Intra-day gas system control is executed in a hierarchical procedure through the deployment of four control actions from various inter-system flexibility providers. The presented analysis illustrates that, based on the selection of control parameters, the activation of flexibility resources from both systems can be steered in such a way as to alleviate linepack deviations, even in cases of severely limited scheduled gas supply. The proposed control framework is tested on the Greek power and gas systems, providing significant insights regarding the activation of the control actions for the real-time gas system balancing in different look-ahead horizons.