The function that you would want to use is sanitize_text_field() which among other things, strips all HTML tags, and remove line breaks, tabs and extra white space.
// This worked fine for me - I got /directory/subdirectory/
$value = sanitize_text_field( '/directory/subdirectory/' );
// And so does this one - I got /thisfile.min.js
$value = sanitize_text_field( '/thisfile.min.js' );
// Even with an actual form input (e.g. <input name="my_field">),
// the function worked just fine (using the same data as above).
$value = sanitize_text_field( $_POST['my_field'] );
So have you already tried that function and are you sure it stripped the /? How about the other similar functions listed here?