The high temperature of the air in power generation gas-turbine cycles involving natural gas (mainly methane) oxidation accounts for the utilization of ion-conductive membranes within solid oxide fuel cells (SOFCs) and membrane reactors (MRs). In SOFCs, the electricity is directly derived from the chemical exergy of methane (SOFCs with internal methane reforming are considered here). Within a membrane reactor (MR), which is considered a substitute for combustion chambers in traditional gas-turbine units, the ion-conductive membranes separate oxygen from air and allow the flow of the hot combustion products (carbon dioxide and steam) to be separated from air. It permits the use of combustion products which are not diluted in nitrogen in the process of methane conversion into hydrogen. A modified gas-turbine cycle that includes a SOFC stack, an MR (instead of a traditional combustion chamber), and a catalytic reactor to convert methane to hydrogen is proposed. An exergy analysis of the proposed system is conducted to evaluate its exergy efficiency and the exergy losses for the processes occurring within the system. It is shown that, in comparison to the traditional gas-turbine cycle, there is a significant reduction (more than three times) in the exergy losses for the most irreversible process occurring in the system, natural gas combustion. It is also found that the proposed cogeneration scheme, including both power generation and the industrial catalytic conversion of methane to hydrogen, permits improved efficiencies for both technologies. The efficiency of this cogeneration, as well as the reduction in exergy losses, is demonstrated by the following observation: if the value of energy (exergy) efficiency of hydrogen production is considered equal to that for a traditional process, the corresponding thermal (energy) efficiency for electricity generation would reach values of 80–96% depending on the efficiency of a SOFC stack. The combined SOFC and MR application also eliminates the possibility of toxic nitrogen oxides formation and, at the same time, makes carbon dioxide removal from flue gases feasible (due to its high concentration). The development of the proposed technology is especially important, within the context of the hydrogen economy, if the produced hydrogen is used as a fuel for fuel cell vehicles.