Add a query string to the end of the theme's style.css when WordPress is loaded.

cssversioner.php

<?php

/**
 * Add a filter to stylesheet_uri, which appends a query string to it automagically.
 */

add_filter('stylesheet_uri', function($css) {
  $parts = parse_url($css);
  $file = $_SERVER['DOCUMENT_ROOT'] . $parts['path'];

  if (file_exists($file)) {
    $query = array();
    if (isset($parts['query'])) {
      parse_str($parts['query'], $query);
    }   
    $query['ver'] = filemtime($file);
    $querystr = http_build_query($query, null, '&amp;');

    $user = isset($parts['user']) ? $parts['user'] : ''; 
    $pass = isset($parts['pass']) ? $parts['pass'] : ''; 
    $userpass = (!empty($user) || !empty($pass)) ? $user.'@'.$pass : ''; 
    $css = "{$parts['scheme']}://{$userpass}{$parts['host']}{$parts['path']}?{$querystr}";
    echo $css;
  }
});