This will usually happen if you are trying to do a PUT or a DELETE and you don't have the API configured to allow it.

This post shows you how to solve this 405 error and get it working properly.

The easiest way to edit the ExtensionlessUrlHandler-Integrated-4.0 line in the web.config file of the Web API project

  <system.webServer>
    <handlers>
      <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
      <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
    </handlers>
  </system.webServer>

I hope this helps. Special thanks to Richard for finding this solution.