What will be destroyed? The Moon or the Death Star? The situation:
Suppose Death Star passes by the Earth towards the Moon (near speed of light) and fires its laser (starting from the center of Death Star) against the Moon. If the laser gets to the Moon, it destroys it, and Death Star passes through. Otherwise (if the laser is still inside the Death Star), Death Star crashes against the Moon.
How is the situation viewed from the Earth:
Since Death Star is traveling almost at the speed of light, when the laser gets to the Moon, the surface of the Death Star is already there. So Death Star crashes.
How is the situation viewed from the Death Star:
The Moon is traveling towards us almost at the speed of light. It will take about one second to get here, so the laser has enough time to leave Death Star. The moon will be destroyed.
The question:
What will be destroyed? The Moon or the Death Star?
 A: 
What will be destroyed? The Moon or the Death Star?

Both.
You are ignoring length contraction. At a "mere" 0.90 c, the 385000 km between the Earth and the Moon is length contracted to 168000 km. The perceived distance grows ever smaller with increased speed: 54000 km at 0.99 c, 17000 km at 0.999 c, 5450 km at 0.9999 c.
You are also ignoring human reaction time and the even slower response time of a militaristic organization such as the one that runs the Death Star. Presumably, the Earth blocked the Death Star's view of the Moon. (If not, they should have blasted the Moon from a much longer distance.) It will take some time for the Death Star crew to see the Moon, even longer for the command to be given and a response made. Even at a mere 0.9 c, the length-contracted distance between the Earth and Moon means that they're all dead by the time the command is given.  (They'll see the Moon 0.56 seconds before they hit it.) At 0.9999 c, they have 18 milliseconds.
Even if they do manage to get that shot off, it probably doesn't save the ship. The planet Alderaan didn't vanish when the Death Star attacked it. The Death Star instead made it explode. Alderaan became chunks of Alderaan. Even one small chunk of the Moon moving at a relativistic speed (from the perspective of the Death Star) would have more than enough kinetic energy to destroy the Death Star.
If they don't manage to get the shot off, the Moon is still going to be destroyed. Assuming the Death Star's average density is about the same as that of air at standard temperature and pressure, the impact of a 150 km radius Death Star moving at 0.99 c with the Moon would release 1034 joules, and in a second or so. That's more than enough energy to destroy the Moon. That's about how much energy the Sun produces in entire year.
