The normal phasings are "finished my master's [degree]", "obtained my/a master's [degree]", or "have my/a master's [degree]". The word degree is optional in all phrasings and should be used where it sounds better. It is frequently omitted when naming the subject (i.e. I hear "master's in engineering" more often than "master's degree in engineering"), but including it still sounds completely normal.
As for "defending the dissertation", that is both meaningful and correct, but it is not terribly common to use the word the in that phrase rather than a pronoun. You more often see "defending his/her/my/your disertation", even when context has already established which dissertation is being discussed.