Chrome doesn't delete session cookies
This can be caused by having Chrome set to Continue where you left off.
Further reading
I just had the same problem with a cookie which was set to expire on "Browsing session end".
Unfortunately it did not so I played a bit with the settings of the browser.
Turned out that the feature that remembers the opened tabs when the browser is closed was the root of the problem. (The feature is named "On startup" - "Continue where I left off". At least on the current version of Chrome).
This also happens with Opera and Firefox.
I just had this issue. I noticed that even after I closed my browser I had many Chrome processes running. Turns out these were each from my Chrome extension.
Under advanced settings I unchecked 'Continue running background apps when Google Chrome is closed' and my session cookies started working as they should.
Still a pain in the rear for all of us developers that have been coding expecting that session cookies would get cleared when the user is done browsing.