This is possible. You can use template_preprocess_page in your custom module to whack Drupal's default CSS:
function mymodule_preprocess_page(&$vars) {
// Get CSS array and remove selected stylesheets
$css = drupal_add_css();
unset($css['all']['module']['modules/system/defaults.css']);
// Override Drupal styles
$vars['styles'] = drupal_get_css($css);
}
Because of Drupal's architecture, you'll almost certainly need to set the module weight to something greater than your other installed modules to ensure that your preprocessor runs last and no other module will be adding CSS.