Mark a Commerce Order as Paid, via a custom plugin Checkout / Payment is being handled entirely by a 3rd Party. What would be the correct process to mark an Order as Paid via a custom Craft plugin? 

The Order will already have been converted from a Cart to an Order and assigned an Order Status via a previous step.

The 3rd Party will be calling a series of API endpoints exposed by a custom Craft plugin to process the various steps of the Order. 

EDIT: For future reference, base on Luke's answer below, this is the code I got to work...

$orderNumber = "XXXX-XXXX-XXXX-XXXX";

$order = craft()-&gt;commerce_orders-&gt;getOrderByNumber($orderNumber);

$transaction = craft()-&gt;commerce_transactions-&gt;createTransaction($order);

$transaction-&gt;status = Commerce_TransactionRecord::STATUS_SUCCESS;
$transaction-&gt;type = Commerce_TransactionRecord::TYPE_PURCHASE;

craft()-&gt;commerce_transactions-&gt;saveTransaction($transaction);

craft()-&gt;commerce_orders-&gt;updateOrderPaidTotal($order);
Your best bet would be to create a transaction for the full amount:

$transaction = Plugin::getInstance()-&gt;transactions-&gt;createTransaction($order);
$transaction-&gt;status = \craft\commerce\records\Transaction::STATUS_SUCCESS;
$transaction-&gt;type = \craft\commerce\records\Transaction::TYPE_PURCHASE;
Plugin::getInstance()-&gt;transactions-&gt;saveTransaction($transaction);
$order-&gt;updateOrderPaidTotal();