Delay-sensitive Internet traffic, such as live streaming video, voice over IP, and multimedia teleconferencing, requires low end-to-end delay in order to maintain its interactive and streaming nature. In recent years, the popularity of delay-sensitive applications has been rapidly growing. This paper provides a protocol that minimizes the end-to-end delay experienced by inelastic traffic. We take a known convex optimization formulation of the problem and use an optimization decomposition to derive a simple distributed protocol that provably converges to the optimum. Through the use of multipath routing, our protocol can achieve optimal load balancing as well as increased robustness. By carrying out packet level simulations with realistic topologies, feedback delays, link capacities, and traffic loads, we show that our distributed protocol is adaptive and robust. Our results demonstrate that the protocol performs significantly better than other techniques such as shortest path routing or equal splitting among multiple paths.