We consider the QoS-aware Multicommodity Flow problem, a natural generalization of the weighted multicommodity flow problem where the demands and commodity values are elastic to the Quality-of- Service characteristics of the underlying network. The problem is fundamental in transportation planning and also has important applications beyond the transportation domain. We provide a FPTAS for the QoSaware Multicommodity Flow problem by building upon a Lagrangian relaxation method and a recent FPTAS for the non-additive shortest path problem.