In Joomla when you call a controller's save() method (usually it is a FormController if it has a save() method) the controller's save() method calls the corresponding Model right away and the validate() and save() methods of that Model. Thus when you want to save something using a component's save() method it is better and more straightforward to call the component's Model and the save() method of that Model directly (not via a controller).
In my comment I've already placed an older link about how to include any Model (and it's methods) in your extension, but now here I am giving an updated version of that:
On the top of your file include:
I'm just giving an example of calling the Banner admin Model from com_banners extension of Joomla:
BaseDatabaseModel::addIncludePath($path = JPATH_ADMINISTRATOR . '/components/com_banners/models', $prefix = '');
$myModel = BaseDatabaseModel::getInstance('Banner', 'BannersModel');
$myModel->save($data); // you can save your data
Link to BaseDatabaseModel Class
If you insist on using the Controller then you have to load the Controller class with the help of JLoader class of Joomla. As an example I'm loading the Banner Controller here (from the admin side of Joomla):
JLoader::register("Banner", JPATH_ADMINISTRATOR . '/components/com_banners/controllers/banner.php');
$myControllerClass = new BannersControllerBanner;
// $methods = get_class_methods($myControllerClass); // to check the available methods
Depending on where you load this class, you can use this with HTMLHelper class methods in your layouts (template files) too.