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Extract and dump a DOM node (and its children) in PHP
<?php
  /* $content actually comes from WP function get_the_content() */
  $content = '<p>some text and so forth that I don't care about...</p> <div class="the-wanted-element"><p>I WANT THIS DIV!!!</p></div>';
?>

$content = '<p>some text and so forth that I don't care about...</p> <div class="the-wanted-element"><p>I WANT THIS DIV!!!</p></div>';

$dom = new DomDocument;
$dom->loadHtml($content);

$xpath = new DomXpath($dom);
$contents = '';
foreach ($xpath->query('//div[@class="the-wanted-element"]') as $node) {
  $contents = $dom->saveXml($node);
  break;
}

echo $contents;

$content = '<p>some text and so forth that I don't care about...</p> <div class="the-wanted-element"><p>I WANT THIS DIV!!!</p></div>';

$dom = new DomDocument;
$dom->loadHtml($content);

$xpath = new DomXpath($dom);
foreach ($xpath->query('//div[@class="the-wanted-element"]') as $node) {
  $node->parentNode->removeChild($node);
  break;
}

$contents = '';
foreach ($xpath->query('//body/*') as $node) {
    $contents .= $dom->saveXml($node);
}

echo $contents;