A system includes a gas turbine engine that includes a combustor section having one or more combustors configured to generate combustion products, a turbine section having one or more turbine stages between an upstream end and a downstream end, an exhaust section disposed downstream from the downstream end of the turbine section, and a fluid supply system coupled to the exhaust section. The one or more turbine stages are driven by the combustion products. The exhaust section has an exhaust passage configured to receive the combustion products as an exhaust gas. The fluid supply system is configured to route a cooling gas to the exhaust section. The cooling gas has a temperature lower than the exhaust gas. The cooling gas includes an extracted exhaust gas, a gas separated from the extracted exhaust gas, carbon dioxide, carbon monoxide, nitrogen oxides, or a combination thereof.