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Take a look at the Template Hierarchy section of the Codex that concerns Custom Post types.

1. archive-{post_type}.php - If the post type were product, WordPress would look for archive-product.php.
2. archive.php
3. index.php

What you are describing is built in, down to the file naming pattern-- archive-cpt.php

To load the same template for all CPT archives use:

function not_post_archive_wpse_107931($template) {
  if (is_post_type_archive()) {
    $template = get_stylesheet_directory_uri().'/archive-cpt.php';
  }
  return $template;
}
add_filter('template_include','not_post_archive_wpse_107931');

That will hijack all CPT archives so I would be very careful with it. It could lead to great frustration if someone can't figure out why a CPT archive is not loading the expected template.

Barely tested. Possibly buggy. Caveat emptor. No refunds.