Varnish/Drupal caching for logged in users

Question

Problem

Varnish is caching pages when I go onto my website anonymously.

I have "Cache Pages for anonymous users" turned on so this is expected.

But when I login it continues to load the cached page for the pages I visited when I was logged out, so the user specific menu never shows and the login button continues to show even though I'm logged in.

I though Varnish didn't work for logged in users and I'm using the default drupal caching system which I thought didn't work for logged in users? How can I prevent this?

Server Set-Up

LEMP Stack - Ubuntu Server 12.04

APC + Memcached

Varnish

Drupal Caching Settings:

Cache pages for anonymous users -> enabled

Minimum cache lifetime -> 0

Expiration of cached pages -> 6 hours

Varnish Module Settings:

Varnish cache clearing: Drupal default

I installed Varnish using this guide and this Varnish config file: http://andrewdunkle.com/how-install-varnish-drupal-7

Browser Caching

I disabled browser caching in Firefox and I still get Varnish cache hits on my homepage however now the menu loads correctly. What does this imply?