Not tested, but try to do a post request via ajax. The file you call via ajax should look something like this:
require( dirname(__FILE__) . '/wp-load.php' );
$user = filter_input( INPUT_POST, FILTER_SANITIZE_STRING );
$pass = filter_input( INPUT_POST, FILTER_SANITIZE_STRING );
is_wp_error(
wp_signon(
array(
'user_login'=>$user,
'user_password'=>$pass
)
)
) ? die( -1 ) : die( 0 );
Put this code in a file (e.g. ajax-login.php) and call it vai ajax. Then WordPress should set a cookie to grant access.