The TOS Kirk earned his promotion to master of the Enterprise after proving himself as the captain of a lesser ship.
Source? Because nowhere in the TOS episodes or movies is such a command mentioned. And yes I know about Dehner's line from Where No Man...
and find it vague at best because the Enterprise could have been the command she mentions.
In fact that was most likely the intent of the line, explaining how Kirk happens to have his best friend under his command on the Enterprise. So the pilot pretty much establishes the Enterprise as his first command.