Recent technological evolution in the Internet of Things (IoT) age supports better solutions to magnify the management of the power quality and reliability concerns, and imposes the measures of a smart grid. In smart grid environment, a smart meter needs to securely access the services from a service provider via insecure channel. However, since the communication is via public channel, it imposes various security threats by an adversary. To deal with this, in this article we design a new anonymous signature-based authenticated key exchange scheme for IoT-enabled smart grid environment, called AAS-IoTSG. The dynamic smart meter addition phase is also permissible in AAS-IoTSG after initial deployment. The security of AAS-IoTSG has been tested rigorously using formal security analysis under the real-or-random (ROR) model which is one of the broadly-accepted standard random oracle models, formal security verification under the broadly-used automated validation of Internet security protocols and applications (AVISPA) tool and also using informal security analysis. Finally, an exhaustive comparative study unveils that AAS-IoTSG supports better security and functionality features and requires less communication and computation overheads as compared to the existing state-of-art authentication mechanisms in smart grid systems.