Most atomic clocks are actually not used to keep time the way a clock on the wall keeps time. They're most useful in measuring the amount of time that has elapsed in a laboratory.
However, those that do keep time (by measuring the amount of time that has passed since some established reference point) are not adjusted for Daylight Savings Time. Daylight Savings Time has no physical analogue, it's simply a seasonal change we make in the definition of what the numbers on a clock mean in terms of time. Because of this and time zone adjustments, reference times like those provided by atomic clocks are generally given in Greenwich Mean Time (alternately called Universal Coordinated Time or Zulu Time). This corresponds to the time zone containing Greenwich, England and is not modified by Daylight Savings Time.
To answer this question, let's first think about how the atomic clock works. Actually, let's start with how any clock works.
Judson Powers, B.S., Physics Grad Student, Cornell, Ithaca, NY
Basically, a clock depends on something that repeats a pattern at a very regular rate. For big old grandfather clocks, it's a pendulum that swings back and forth with a very regular period (so that you can tell exactly how long it takes for one swing). For most watches, it's a crystal that, when you apply power to it, vibrates very rapidly, with a very stable frequency. For the atomic clock, it's atoms (usually cesium-133) which emit a very precise frequency of light when they fall from an excited state to a ground state.
This thing that oscillates is the heart of the clock, and then for the rest of it, you just need some sort of device that counts the oscillations and converts a number of oscillations into a time that you can understand / recognize. In the grandfather clock, it's a system of gears that converts the pendulum period into appropriate speeds to move the second hand, minute hand, and hour hand. In digital watches, it's simply electronics which count the number of oscillations of the crystal and divide by an appropriate number to get seconds, minutes, etc. In an atomic clock, it's the same - it just takes some fancy optics and electronics to convert the frequency of the light into a time signal that's slow enough for us to use it.
Now daylight savings time is really something we impose on our clocks; when you switch to DST, an hour of time doesn't really INSTANTANEOUSLY pass - we just set our clocks forward. You don't stop the clock and wait, or speed up the oscillations to "catch up" (well, I suppose you could, but it would be INSANELY hard). Most clocks simply have a built-in function that allows you to adjust what time they read without having to rely on the internal timekeeper. In analog clocks, it's usually a knob that allows you to turn the hands yourself. In digital watches, and the atomic clocks, there's simply a layer of electronics between the part that counts the vibrations of the crystal / atom and the part that tells you what time it is, and this layer only has the effect that it can change the digits that you see.
Depending on whether or not I read the question right, this could be a long involved response for a very simple answer, but I'm hoping that it can potentially de-mystify some of the internal workings of our time-keeping devices.
For more information on atomic clocks, there's a bunch of good stuff at http://tycho.usno.navy.mil/cesium.html
Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN