Security testing verifies that the data and the resources of software systems are protected from attackers. Unfortunately, it suffers from the oracle problem, which refers to the challenge, given an input for a system, of distinguishing correct from incorrect behavior. In many situations where potential vulnerabilities are tested, a test oracle may not exist, or it might be impractical due to the many inputs for which specific oracles have to be defined. In this paper, we propose a metamorphic testing approach that alleviates the oracle problem in security testing. It enables engineers to specify metamorphic relations (MRs) that capture security properties of the system. Such MRs are then used to automate testing and detect vulnerabilities. We provide a catalog of 22 system-agnostic MRs to automate security testing in Web systems. Our approach targets 39% of the OWASP security testing activities not automated by state-of-the-art techniques. It automatically detected 10 out of 12 vulnerabilities affecting two widely used systems, one commercial and the other open source (Jenkins).