The Death Star used a superlaser - a beefed-up, continuous-beam version of a turbolaser. Like a conventional turbolaser or a handheld blaster, a superlaser isn't actually a beam of coherent light; that would tend to spread out over long distances, if it could even be built to deliver the kind of energies involved. Rather, it emits a long, self-contained "bolt" of stimulated plasma, which generates its own magnetic containment field.
The twisted genius of the Death Star was that the initial plasma, to be accelerated through magnetic firing tubes in a conventional bolt and then combined into the final, staggeringly powerful planet-killer shot, was not produced using excited Tibanna Gas (as in a conventional blaster.) Rather, it made use of vast Hypermatter fusion reactors; four-dimensional exotic matter that extended into hyperspace. This, obviously, can produce exponentially higher energy yields than could be contained in the same volume of conventional matter via E=MC2; in much the same way a sphere is larger than a circle; which is why they were originally developed. But more importantly for the Death Star project, the high-energy collisions between these hypermatter particles produces short-lived four-dimensional singularities; microscopic portals into hyperspace.
When the hypermatter plasma from the reactor encounters regular matter, the effect is immediate; vast quantities of matter are pulled into hyperspace, as the remaining material is superheated by the plasma and bombarded with high-energy radiation from the singularities.
This is why the destruction of Alderaan produced a large, rapidly-moving debris field - which would have required orders of magnitude more plasma than a conventional weapon could provide - yet one surprisingly small, considering the planets size, with no threat to the Death Star or observing Imperial ships. It also explains the lack of dangerous debris after the station itself was destroyed by the damaged superweapon; an explosion with much sound and fury, but surprisingly little teeth for the rebel forces to contend with.
So; on to your question.
The Death Star superlaser was incapable of destroying a gas-giant; it simply didn't produce enough energy to break it apart. However, that is not to say it would have no effect. Had it penetrated to the core, as you hypothesized, it would indeed have destroyed the center of the planet; resulting in a truly massive implosion.
However, I suspect would instead create a staggeringly large explosion in the upper atmosphere; taking a chunk of the planet with it the size of Alderaan. The cataclysmic storms produced would involve literally unimaginable forces; the giant's diameter would actually reduce to a degree visible to the naked eye. I am not familiar with the chemical composition of the Yavin gas giant, but as Tibanna gas is often plentiful in gaseous planets, it is even possible the atmosphere would ignite like a a gigantic blaster tube, flinging vast moon-sized chunks of undirected plasma into spacearound the inactive site.
However, unfortunately, the Death Star's tactical position would not be improved; as it required 24 hours to recharge.
This might not be canon these days.]