You are confusing the title of captain with the rank of Captain. The difference is referenced in the opening paragraph of the Memory Alpha article Captain:
Captain is a military rank, the equivalent of which was used by the service organizations of many civilizations. The title of captain was often used by vessel commanders and, as a naval rank, in many Earth navies and Starfleet, was above commander. As an infantry rank, captains ranked subordinate to a major and above a first lieutenant. The abbreviation for captain was capt.
While that can be one and the same (especially in the Star Trek universe), the title of captain is given to the commanding officer of a vessel, whereas the rank is given to an individual who was promoted from the previous rank (in Starfleet, it would be a promotion from Commander, as Sisko was).
For example, in Star Trek V: The Final Frontier and Star Trek VI: The Undiscovered Country, Captain Kirk held the rank of Captain and was also given the title of captain as commanding officer of the USS Enterprise. At the same time, Captain Spock was not captain of the ship (despite having the rank of Captain), but held the position of First Officer. Additionally, as user16696 reminded me, Scotty also held the rank of Captain by the time of the last movies.
Similarly, in Star Trek: The Motion Picture, Admiral Kirk was given command of the USS Enterprise so was addressed as "captain" throughout the movie because he was given that title as the commanding officer of the Enterprise.
The Defiant itself also illustrates this difference, although the distinction was not openly addressed until later in the series. When he brought the Defiant back to DS9, Sisko still held the rank of Commander, but when he was in command of the Defiant, he technically held the position of captain (although no one addressed him as such). Then, early in Season 6 of DS9 ("Behind the Lines"), Lt Commander Dax was assigned as the commanding officer of the Defiant, so was given the title of captain, and she was actually referred to as "captain" by the crew while she was in command. In fact, there is a dialog between O'Brien and Cadet Nog about this exact situation:
O'Brien: "It's an old naval tradition. Whoever's in command of a ship, regardless of rank, is referred to as 'captain'."
Nog: "You mean if I had to take command, I would be called 'captain,' too?"
O'Brien: "Cadet, by the time you took command, there'd be nobody left to call you anything."
So to address your question more directly, the reason Sisko was promoted to Captain was because he earned the increase in rank through his service to Starfleet and the Federation. It is not related to the Defiant at all, simply because he already held the title of captain every time he commanded the ship.