I know this is old but maybe someone comes here looking for answer like me, this is tested on Drupal 8.

I found this [issue][1] and after looking at last [patch][2], I implemented :

    $nodeDuplicate = $node->createDuplicate();

    foreach($nodeDuplicate->field_paragraphs as $field){
        $field->entity = $field->entity->createDuplicate();
    }

    $nodeDuplicate->save();

And I got duplicate node with duplicated paragraphs fields. If you look at patch you will see how to add button to Drupal etc.

Just a note `field_paragraphs` is name of my field in content type, change it with your own.

I didn't have any problem doing it this way, maybe someone has a better way but I did not find it, also at time of writing this there is still no ported modules for cloning nodes in Drupal 8.
  [1]: https://www.drupal.org/node/2829316
  [2]: https://www.drupal.org/files/issues/implement_duplicate-2829316-53.patch