An asynchronous operation cannot be started at this time Exception occurs on calling WebService?

Make sure that your Controller method returns an async Task.

public class ServiceController : Controller 
{
    public async Task<ActionResult> Index()
    {       
        var service = new Service();
        await service.CallMethodAsync();    
        return View();
    }
}

Basically, the documentation is written in a way where they believe you are only using ASP.NET WebForms, however obviously you can use this in MVC applications too, so their documentation needs to be updated.