How to get comment_ID by post_ID in wordpress

Use get_comments( $args ) to get all the comments for a post.

$args= array(
    'author_email' => '',
    'ID' => '',
    'karma' => '',
    'number' => '',
    'offset' => '',
    'orderby' => '',
    'order' => 'DESC',
    'parent' => '',
    'post_id' => 0,
    'post_author' => '',
    'post_name' => '',
    'post_parent' => '',
    'post_status' => '',
    'post_type' => '',
    'status' => '',
    'type' => '',
    'user_id' => '',
    'search' => '',
    'count' => false,
    'meta_key' => '',
    'meta_value' => '',
    'meta_query' => '');

You can set post_id here and get all comments related to that post. It is same as your query.

$comments = get_comments(array('post_id'=> 'your post id'));
$comment_ids = array();
foreach($comments as $comment):
    $comment_ids[] = $comment->comment_ID;
endforeach;
print_r(comment_ids);

For reference http://codex.wordpress.org/Function_Reference/get_comments