Controller代码如下：
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Cors;
using Microsoft.AspNetCore.Mvc;
using Microsoft.AspNetCore.Http;

namespace WebApplication2.Controllers
{
    [Route("api/[controller]")]
    [EnableCors("any")] 
    [ApiController]
    public class TestController : Controller
    {

        // GET api/test/5
        [HttpGet("{id}")]
        [EnableCors("any")] 
        public string  Get(int id)
        {
            HttpContext.Session.SetString("name", "中文");
            string result = HttpContext.Session.GetString("name");
            return result.ToString();
        }

        // GET api/test
        [HttpGet]
        public string Get()
        {
            try
            {
                string result = HttpContext.Session.GetString("name");
                return "name:"+result.ToString();
            }
            catch (Exception ex)
            {
                return ex.ToString();
            }
        }

    }
}

Startup代码如下：
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Builder;
using Microsoft.AspNetCore.Hosting;
using Microsoft.AspNetCore.Http;
using Microsoft.AspNetCore.HttpsPolicy;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Extensions.Configuration;
using Microsoft.Extensions.DependencyInjection;
using Microsoft.Extensions.Logging;
using Microsoft.Extensions.Options;

namespace WebApplication2
{
    public class Startup
    {
        public Startup(IConfiguration configuration)
        {
            Configuration = configuration;
        }

        public IConfiguration Configuration { get; }

        // This method gets called by the runtime. Use this method to add services to the container.
        public void ConfigureServices(IServiceCollection services)
        {
            //Session 保存到内存
            services.AddDistributedMemoryCache();
            services.AddSession();

            //配置跨域处理
            services.AddCors(options =>
            {
                options.AddPolicy("any", builder =>
                {
                    builder.AllowAnyOrigin() //允许任何来源的主机访问
                    .AllowAnyMethod()
                    .AllowAnyHeader()
                    .AllowCredentials();//指定处理cookie
                });
            });

            services.AddMvc().SetCompatibilityVersion(CompatibilityVersion.Version_2_1);

        }



        // This method gets called by the runtime. Use this method to configure the HTTP request pipeline.
        public void Configure(IApplicationBuilder app, IHostingEnvironment env)
        {
            if (env.IsDevelopment())
            {
                app.UseDeveloperExceptionPage();
            }
            else
            {
                app.UseHsts();
            }
            app.UseSession();
            app.UseHttpsRedirection();
            app.UseMvc();
        }
    }
}