public void Configuration(IAppBuilder app)
{
    app.SetDefaultSignInAsAuthenticationType(CookieAuthenticationDefaults.AuthenticationType);
    app.UseWsFederationAuthentication(
        new WsFederationAuthenticationOptions
        {
            Wtrealm = realm,
            MetadataAddress = adfsMetadata
        });
    app.UseCookieAuthentication(new CookieAuthenticationOptions
    {
        AuthenticationType = WsFederationAuthenticationDefaults.AuthenticationType
    });
}

using System;
using System.IdentityModel.Protocols.WSTrust;
using System.IdentityModel.Tokens;
using System.Net;
using System.Net.Http;
using System.ServiceModel;
using System.ServiceModel.Security;
using Thinktecture.IdentityModel.Extensions;
using Thinktecture.IdentityModel.WSTrust;

namespace ConsoleApplication1
{
    class Program
    {
        private const string UserName     = "USERNAME";
        private const string Password     = "PASSWORD";
        private const string Domain       = "DOMAIN";
        private const string ADFSEndpoint = "ADFS ENDPOINT";
        private const string ApiBaseUri   = "THE API";
        private const string ApiEndPoint  = "AN ENDPOINT";

        static void Main(string[] args)
        {
            SecurityToken token = RequestSecurityToken(); // Obtain security token from ADFS.
            CallApi(token);                               // Call api.
            Console.ReadKey();                            // Stop console from closing
        }

        private static SecurityToken RequestSecurityToken()
        {
            var trustChannelFactory =
                new WSTrustChannelFactory(new UserNameWSTrustBinding(SecurityMode.TransportWithMessageCredential),
                    new EndpointAddress(new Uri(ADFSEndpoint)))
                {
                    TrustVersion = TrustVersion.WSTrust13,
                    Credentials = { UserName = { UserName = UserName + "@" + Domain, Password = Password } },
                };

            var requestSecurityToken = new RequestSecurityToken
            {
                RequestType = RequestTypes.Issue,
                KeyType = KeyTypes.Bearer,
                AppliesTo = new EndpointReference(ApiBaseUri)
            };

            RequestSecurityTokenResponse response;
            var securityToken = trustChannelFactory.CreateChannel().Issue(requestSecurityToken, out response);

            return securityToken;
        }

        private static async void CallApi(SecurityToken securityToken)
        {
            using (var handler = new HttpClientHandler { CookieContainer = new CookieContainer() })
            {
                using (var client = new HttpClient(handler))
                {
                    handler.CookieContainer.MaxCookieSize = 8000; // Trying to make sure I can fit it in the cookie

                    var cookie = new Cookie {
                        Name = "FedAuth",
                        Value = Base64Encode(securityToken.ToTokenXmlString()),
                        HttpOnly = true,
                        Secure = true
                    };
                    handler.CookieContainer.Add(new Uri(ApiBaseUri), cookie);
                    var response = client.GetAsync(new Uri(ApiBaseUri + ApiEndPoint)).Result;
                    string result = await response.Content.ReadAsStringAsync();
                    Console.WriteLine(result);
                }
            }
        }

        public static string Base64Encode(string plainText)
        {
            var plainTextBytes = System.Text.Encoding.UTF8.GetBytes(plainText);
            return System.Convert.ToBase64String(plainTextBytes);
        }
    }
}