In the original Space Seed, it is stated:
In 1993, a group of these young supermen did seize power simultaneously in over forty nations.
We know later that some of these nations were in Asia and the Middle East.
There are many historical examples of coup d'etats where terrorism wasn't used. Wikipedia describes the coup d'etat and lists both successful and unsuccessful ones. A number of coup d'etats were orchestrated by the military, so it's conceivable to me that people like Khan were able to convince the military leaders of weak nations, as one could find in Asia and Middle East, to join his cause and overthrow an existing government.
Khan wasn't depicted as a lovable dictator. Yet, he wasn't depicted as a murderous dictator either. This interpretation came up later.
(Space Seed is quite clear on Khan not being a murderous dictator when Scotty said:
There were no massacres under his rule.