According to Wookieepedia, the Death Star 1's diameter is 160 km, while Alderaan's is 12,000 km. After shooting Alderaan one can assume there should be large and very fast chunks of the planet flying towards the Death Star. To put these dimensions in context, a small asteroid over 100 m caused the Tunguska event devastating 2000 square kilometres. How did the Space Station manage this? Can its shields deflect large asteroids?
These calculations are approximate:
12000km diameter planet = 1.1e6 square km of debris.
Death Star max range is 2000000km to 240000000km.
Effective volume is 2.5e13 to 1.8e17 so around 2 parts per 1e7 to 1e11.
Given the above, there is a good chance of avoiding everything, if anything got close they have plenty of time to respond. Recommended operational distance is 6 light seconds, suppose that matter flies away at 6x the speed of sound they have 11 days to respond to incoming debris.
This is a simple matter for a fleet of Tie fighters and a fully operational Death Star.
Information from wookiepedia. http://starwars.wikia.com/wiki/Superlaser
According to C-canon ("Death Star" book by by Michael Reaves and Steve Perry), most of the mass of the targeted planed is NOT exploded outward as matter:
It took no more than an instant. Tenn knew that the beam's total destructive power was much bigger than matter-energy conversions limited to realspace. At full charge, the hyper-matter reactor provided a superluminal "boost" that caused much of the planet's mass to be shifted immediately into hyperspace. As a result, Alderaan exploded into a fiery ball of eye-smiting light almost instantaneously, and a planar ring of energy reflux-the "shadow" of a hyperspatial ripple-spread rapidly outward.
In other words,
There was no mass of debris for Death Star to worry about avoiding. Instead of a mass of planets, what was left was light and some chunks (as can be seen in the movie) that look like a lot but - at a significant distance - constitute much less of a threat of a hit.
Most of the explosive energy was converted into hyperspace and into light - less was left to impart enough kinetic energy to those chunks to accelerate far.
In the Star Wars Universe, what shields can block is determined by how many shield projectors you have and how much power you give them. The Death Star is described as having many, many projectors and a LOT of power to work with. It may very well be able to deflect a large chunk of planet.
It wouldn't need to, however.
As you can see in Empire Strikes Back, Star Destroyers are more than capable of destroying even large asteroids with their turbolasers. Star Destroyers aren't in any danger of damage from a typical Star Wars asteroid field, unless your commanding officer has ordered you to drop shields so you can carry on a full-holo communication, of course.
In any case, the strength of the Death Star's shields isn't really at question - it was covered with thousands of turbolasers. This Wookiepedia article lists the first Death Star as having over 10,000 turbolasers and 2,500 laser canons. It also gives the effective range of the superweapon as 47,060,000 kilometers. Between all the turbolasers ready to superheat any incoming rocks into rapidly-dispersing plasma, extreme range meaning they have very few incoming rocks to melt, and shields that could hold off a fleet of star destroyers pretty much indefinitely, the Death Star has very little chance of immolating itself upon it's victims' funeral pyres.
The smallest visible debris are over 100 kilometers in diameter and were travelling at thousands of kilometers per second. Debris of this size and speed would be impacting the Death Star with planet shattering energies within seconds of Alderaan being destroyed.
Regarding the Death Star hyperspace boost
Canon novel of A NEW HOPE
"Luke had seen the shattered remains of Alderaan and knew that for those in the incredible battle station the entire moon would present simply another abstract problem in mass-energy conversion."
This over-rules the non-canon Death Star novel which mentions hyperspace mumbo-jumbo technobabble, and tells us the Death Star does indeed use a conventional hypermatter fuel consuming superlaser.
Even in the Expanded Universe the Death Star novel is an outlier as numerous other novels all simply state the Death Stars reactor builds up the energies needed for a planet destroying burst with no mention of this hyperspace stuff, and some such as "BEHIND THE MAGIC" even state the superlaser is a form of up scaled turbolaser weaponry.
Many like to read Death Stars technobbable explanation and conclude the Death Star must be less powerful to achieve this. But even if we went with the wobbly explanation in that book it would only serve to increase the firepower, not lower it, because accelerating something into FASTER THAN LIGHT SPEEDS is obviously more energy consuming than accelerating the planet to SUBLIGHT SPEEDS. Furthermore the Return of the Jedi novel even states accelerating to lightspeed is a prerequisite to jumping to FTL.
They used shields and defense turrets.