Using forms authentication cookie in WPF's browser control via Web API

We have an ASP.NET web application that uses Forms authentication. We also have a WPF application which uses a Web browser control so it can use some pages from the web application. In order to authenticate users in WPF application, I created a Web API post method as follows:

public class LoginController : ApiController
{
    [HttpPost]
    public bool Post(LoginModel login) 
    {
        User user;
        var success = UserRepository.Authenticate(login.UserName, login.Password, out user);
        if (success)
        {
            //FormsAuthentication.SetAuthCookie(login.UserName, true);
            var authCookie = LedgexSession.CreateAuthorizationCookie(login.UserName, true);
            System.Web.HttpContext.Current.Response.SetCookie(authCookie);
            return true;
        }
        return false;
    }

    public class LoginModel
    {
        public string UserName { get; set; }

        public string Password { get; set; }
    }
}

This code works fine, and I am able to hit the Web Api method from postman and from my WPF application and get success as a return value. However, the WPF's web browser control still doesn't get the authenticated cookie, and anytime I open a page in the control, I am taken to the login screen. How can I set its cookie to the authenticated one I get from Web API?