The goal of multi-objective quality-driven service selection (QDSS) is to find service selections for a workflow whose quality-of-service (QoS) values are Pareto-optimal. We consider multiple QoS attributes such as response time, cost, and reliability. A selection is Pareto-optimal if no other selection has better QoS values for some attributes and at least equivalent values for all others. Exact algorithms have been proposed that find all Pareto-optimal selections. They suffer however from exponential complexity. Randomized algorithms scale well but do not offer any formal guarantees on result precision. We present the first approximation scheme for QDSS. It aims at the sweet spot between exact and randomized algorithms: It combines polynomial complexity with formal result precision guarantees. A parameter allows to seamlessly trade result precision against efficiency. We formally analyze complexity and precision guarantees and experimentally compare our algorithm against exact and randomized approaches. Comparing with exact algorithms, our approximation scheme allows to reduce optimization time from hours to seconds. Its approximation error remains below 1.4 percent while randomized algorithms come close to the theoretical maximum.