Well let's consider what we need in order to have an atmosphere: Gravity. You could think of an atmosphere as a world-wide version of tug-of-war. Various forces are pushing the gases out, while gravity is pulling those same gases back in. So we would need sufficient gravity to keep an atmosphere, and in this case, size doesn't matter. We can't judge the gravity exerted by this space station just by the fact that it appears to be the size of a small moon.
The first issue at hand is that the artificial gravity within the Death Star doesn't appear to center around the Death Star. As we can see in Star Wars (Episode 4), the gravity actually appears to be perpendicular to the center of mass. With that, we cannot correlate any internal gravity with its external gravity.
Now we know that the Death Star has thermal exhaust ports to pump the excess heat from the core reactor. Based on the design and appearance of these exhaust ports, we could infer that they likely expel a heated gas. 921
However it wouldn't make much sense to expel a heated gas if the Death Star had the kind of atmosphere that would keep it all in. Essentially, this would cook the Death Star from both inside and outside since there would be no place for this heat to go. This makes me believe that it does not maintain the gravity necessary to maintain such an atmosphere, except....
We have explosions happening on the surface of the Death Star. Depending on how much you wish to attribute to movie magic, we could probably wave this off, but let's assume it's how things actually worked. There was definitely fire resulting from these, and so we would have to assume that there was oxygen to fuel the flame. What makes this even more confusing is the fact that there are several breaches into the interior, where oxygen exists, but there is no vacuum. So either they had a way of containing these very quickly or there was an atmosphere on the exterior sufficient enough for there to be no pressure difference.
The fact that we need specifically shielded airlocks compounds this even more because we wouldn't need them if the Death Star had enough of an atmosphere to make it negligible. More than likely, this atmosphere is either partial (ie, it covers only some of the surface of the Death Star), or is held by forces weak enough for the atmosphere to be expelled at some rate.
Based on these two conflicting pieces, I would say that the Death Star does maintain an atmosphere, just not a very good one. 513