Hi,
In the Microsoft world what you're looking for is called NLB cluster (Network Load Balancing). That's the way I would reach the goal if the Web servers are Miscrosoft IIS servers.
I don't know much about non-microsoft worl but surely DNS round robin will not do the job because when you DNS client cache expires you could be routed to a web server that doesn't know anything about you current session (except if you have a process inside your application that can replicate session contexte between web servers).
When you configure NBL cluster you can do thing so that one client IP address will always be routed to one NLB cluster node.
Have a nice day