Component-based approaches to program synthesis assemble programs from a database of existing components, such as methods provided by an API. In this paper, we present a novel type-directed algorithm for component-based synthesis. The key novelty of our approach is the use of a compact Petri-net representation to model relationships between methods in an API. Given a target method signature S, our approach performs reachability analysis on the underlying Petri-net model to identify sequences of method calls that could be used to synthesize an implementation of S. The programs synthesized by our algorithm are guaranteed to type check and pass all test cases provided by the user. We have implemented this approach in a tool called SyPet, and used it to successfully synthesize real-world programming tasks extracted from on-line forums and existing code repositories. We also compare SyPet with two state-of-the-art synthesis tools, namely InSynth and CodeHint, and demonstrate that SyPet can synthesize more programs in less time. Finally, we compare our approach with an alternative solution based on hypergraphs and demonstrate its advantages.