Man. This was not easy to chase down. The answers up above absolutely do not work unless you give the user’s role very broad “manage_options” capability. I asked (and answered) the Trac issue here: Admin custom option screen not saved unless user manage_options capability
Anyone in the future who might search/read this – the trick is to leverage that new filter capability to register your required capability for the more narrow “manage options” activity for your specific plugin or theme.