UnhandledExceptionFilter.cs

public class UnhandledExceptionFilter : ExceptionFilterAttribute {
    public override void OnException(HttpActionExecutedContext context) {
        Elmah.ErrorLog.GetDefault(HttpContext.Current).Log(new Elmah.Error(context.Exception));
    }
}

WebApiConfig.cs

public static class WebApiConfig {
    public static void Register(HttpConfiguration config) {
        config.Routes.MapHttpRoute(
            name: "DefaultApi",
            routeTemplate: "api/{controller}/{id}",
            defaults: new { id = RouteParameter.Optional }
        );

        config.Filters.Add(new UnhandledExceptionFilter());
    }
}