AppHost.cs

using System;
using Autofac;
using Funq;
using System.Linq;
using System.Net;
using System.Reflection;
using System.Web;
using ServiceStack.ServiceInterface;
using ServiceStack.Common;
using ServiceStack.ServiceInterface.Admin;

namespace MyServices.Local.API
{
    public class AppHost : AppHostHttpListenerBase
    {
        private readonly IContainer _container;

        public static Assembly[] IncludedAssemblies { get; set; }

        public AppHost(IContainer container)
            : base("My API", IncludedAssemblies)
        {
            _container = container;
        }

        public override void Configure(Container container)
        {
            LogManager.LogFactory = new SerilogFactory(Log.Logger);
            var adapter = new AutofacIocAdapter(_container);
            container.Adapter = adapter;
            RequestFilters.Add((req, resp, dto) =>
            {
                var scope = _container.BeginLifetimeScope(MatchingScopeLifetimeTags.RequestLifetimeScopeTag);
                HostContext.Instance.Items["AutofacScope"] = scope;
            });
        }

    }
}

ScopedAutofacIocAdapter.cs

using Autofac;
using ServiceStack.Common;
using ServiceStack.Configuration;

namespace MyServices.Local.API.Base
{
    public class AutofacIocAdapter : IContainerAdapter
    {
        public IContainer Container { get; private set; }

        public AutofacIocAdapter(IContainer container)
        {
            Container = container;
        }

        public T Resolve<T>()
        {
            return !HostContext.Instance.Items.Contains("AutofacScope")
                ? Container.Resolve<T>()
                : ((ILifetimeScope) HostContext.Instance.Items["AutofacScope"]).Resolve<T>();
        }
 
        public T TryResolve<T>()
        {
            T result;
            return Container.TryResolve(out result) ? result : default(T);
        }

    }
}