What is the proper way to structure this sentence?

The original is

The pact brought power; power brought notoriety;/,/ ... and notoriety brought the authorities.

(Some would quibble about the second semi-colon; I'd keep it for balance, in spite of the coordinator. I feel two commas doesn't lend the required gravitas, and two ellipses looks a little unprofessional. Fewer would probably contest the more obvious 'super-comma' usage in your deleted version:)

The pact brought power; power, notoriety; and notoriety, the authorities.

(though I think I'd use an ellipsis in place of the second semicolon to show a heightening of dramatic tension).

Omitting the repeats of the verb is known as gapping.

Dictionary.com has:

gapping [noun] Linguistics.

a rule of transformational grammar by which repeated instances of a verb are deleted from conjoined sentences, as in the deletion of brought from

Mary brought the bread, John the cheese, and Bill the wine.