Network Functions Virtualization (NFV) is a network architecture concept where network functionality is virtualized and separated into multiple building blocks that may connect or be chained together to implement the required services. The main advantages consist of an increase in network flexibility and scalability. Indeed, each part of the service chain can be allocated and reallocated at runtime depending on demand. In this paper, we present and evaluate an energy-aware Game-Theory-based solution for resource allocation of Virtualized Network Functions (VNFs) within NFV environments. We consider each VNF as a player of the problem that competes for the physical network node capacity pool, seeking the minimization of individual cost functions. The physical network nodes dynamically adjust their processing capacity according to the incoming workload, by means of an Adaptive Rate (AR) strategy that aims at minimizing the product of energy consumption and processing delay. On the basis of the result of the nodes’ AR strategy, the VNFs’ resource sharing costs assume a polynomial form in the workflows, which admits a unique Nash Equilibrium (NE). We examine the effect of different (unconstrained and constrained) forms of the nodes’ optimization problem on the equilibrium and compare the power consumption and delay achieved with energy-aware and non-energy-aware strategy profiles.