For some unknown reason we weren't able to rename the remote file using sftp_rename function. Instead we've used the following code:
<?php
$connection = ssh2_connect('localhost', 22);
ssh2_auth_password($connection, 'username', 'password');
$sftp = ssh2_sftp($connection);
$fromSftpUrl = "ssh2.sftp://{$sftp}/{$path_from}";
$toSftpUrl = "ssh2.sftp://{$sftp}/{$path_to}";
$success = rename($fromSftpUrl, $toSftpUrl);
?>