We propose a framework of joint energy sustainability and QoS provisioning for energy harvesting Wireless Mesh Networks (WMN) equipped with battery. We consider end-to-end delay and throughput as QoS parameters in the proposed framework. Our framework extends the concept of virtualization to energy resources of a node and dedicates a virtual battery to each connection independently. More importantly, we present a novel approach for coupling delay and throughput requirements to their equivalent virtual battery specification, providing a trade-off between QoS and energy characteristics of a connection. Furthermore, we propose admission tests based on average and instantaneous energy requirements of a connection in relation to its QoS needs. These admission tests ensure an end-to-end soft delay bound which according to our simulations, is violated by merely 2%. Using our framework we also show the strikingly different energy policies that should be adopted for routing interactive and streaming as well as bursty and constant bit-rate applications over energy harvesting WMNs. Obtaining an upper bound on the amount of energy resources required within any network clique, we show how energy constrained networks can benefit from increasing transmission power or even reduced bit-rates, despite intuition. All our claims and discussions are backed both by simulations and analysis.