After successfully logging in via the dashboard gadget, the JSESSIONID cookie is set with its Path set to the tomcat context path + "/". Contrast this to the atlassian.xsrf.token cookie Path which is set to just the context path (no trailing slash).

After successful login, Chrome redirects to the root of the JIRA installation. Since the root path lacks the trailing slash, the JSESSIONID cookie is not sent (only the atlassian.xsrf.token cookie is sent), which results in JIRA responding with a new, "lout" session ID cookie and displaying the login gadget on the dashboard.

Logging in via the /index.jsp path works as expected.

Tested using Chromium 32.0.1700.102.
JIRA context path set to /jira.