Optimizing the energy consumption of sensor nodes and forming an efficient communication structure among these nodes are the main design challenges in Wireless Sensor Networks (WSNs). Topology control (TC) is a technique through which network controls its nodes for efficient resource utilization. A Connected Dominating Set (CDS) based TC is a competitive approach among the existing methods used for constructing a Virtual Backbone (VB) in WSNs. In order to reduce the communication cost of WSNs, many researchers focus on constructing a Minimum CDS (MCDS). The problem with having a MCDS based TC is that it is possible to lose connection with some part of the network (or the whole network) even if only a single dominator becomes unusable (i.e., dies or does not have enough energy). As a result, prolonging the time interval before the failure of the first dominator node is crucial issue for most CDS based WSNs. This paper tackles with extending this time interval which is called as stable or constant period of a CDS. To handle topology changes due to dominator failure, a new CDS should be constructed which may be intensive in terms of computation and/or energy. This paper proposes an evolutionary method based protocol called Stability Aware Evolutionary CDS (SAECDS) to prolong the stability period of a CDS by formulating a new fitness function defined as the minimization of the total dissipated energy in the network. To the best of our knowledge, this is the first paper that considers prolonging the stability of CDS using an evolutionary algorithm. Extensive simulation results indicate that SAECDS always results in longer stability periods compared to following algorithms: a heuristic based method (Greedy area mCDS) and a genetic algorithm based method (RMCDS-GA).