With the rapid development of information and communication technology (ICT), the demand for flexible and cost-effective network services (NSs) is growing exponentially. Network function virtualization (NFV) based on elastic optical data center interconnections (EO-DCI) can provide flexible and timely NSs. One of the major concerns that draws the attention of researchers is the exponential growth of the energy consumption of the EO-DCI networks. Therefore, it is a practical issue to reduce the energy consumption of service deployment in EO-DCI networks while ensuring service success. In this article, a flexible service provisioning based on virtual network function chain (vNFC) is exploited. Then we first formulate the energy-efficient vNFC deployment (EE-VNFD) problem in EO-DCI networks and propose an Integer Linear Programming (ILP) model of it by considering the four energy consumption components of CPUs, ports, transponders, and amplifiers. To obtain feasible solutions for real-scale problems, we propose an energy-efficient graph reinforced vNFC deployment (EGRD) algorithm based on reinforcement learning (RL) and graph convolutional networks (GCN). The performance of the EGRD algorithm is evaluated in both static and dynamic scenarios. In the static scenario, simulation results show that the EGRD algorithm achieves a near-optimal performance close to the ILP model. In the dynamic scenario, compared with two heuristic algorithms and two leading RL algorithms, the EGRD algorithm significantly reduces energy consumption in the resource-sufficient environment, and also balances energy consumption and blocking probability well in resource-limited environments.