An important evolution of the users’ needs is represented by the on-demand access to the network, storage, and compute resources in order to dynamically match the level of resource consumption with their service requirements. The response of the network providers is to transition to an architecture based on softwarization and cloudification of the network functions. This is the rationale for the deployment of network functions virtualization (NFV) where virtual network functions (VNFs) may be chained together to create network services. Efficient online routing of demand across nodes handling the functions involved in a given service chain is the novel problem that we address in this paper. We provide an original formulation of this problem that includes link and CPU capacity constraints and is based on the construction of an expanded network. We derive the exact mathematical formulation and propose several heuristic algorithms taking into account the main system’s parameters. We conclude by deriving some interesting insights both about the algorithms and the network performance by comparing the heuristics with the exact solutions.