How do you implement a custom submit handler in hook_form_alter()?
According to the example in the documentation it's the same as for Drupal 7:

$form['actions']['submit']['#submit'][] = 'mymodule_upload_enabled_types_submit';


You can find similar (working) logic in 


contact_form_user_form_alter()
editor_form_filter_format_form_alter()
locale_form_language_admin_edit_form_alter()


And a few others besides.