Rome came to be the dominators by the manner in which they constantly assimilated other people's tactics and weapons for themselves. The Roman Gladius was inspired from the Iberian tribes. The same with the Scutum. Chainmail was invented by the Celts, and the Romans assimilated that as well. The excellent Phoenician shipdesign of the Carthaginians were also assimilated.
Their method of operations and doctrines allowed for total flexibility. Everything from their checker-board formations to the autonomous nature of the maniples, and later cohorts.

They weren't necessarily very good at close-combat either. Remember that by the 1st and 2nd Punic war, the Roman legionaires were if anything.. average at best. Of course, by the Marian reforms, their quality changed drastically.
It was everything together that worked to their advantage. As Socrates said, a disorganized rabble is no more an army than a heap of building-materials is a house.