Dragon has Like-Likes and Wizzrobes, and sometimes both creatures will be in the same room, so equip your whistle just in case the Like-Likes get you.  If the Like-Likes eat up your magic shield, leave Dragon and get a new one.

Once inside, always be on the lookout for Wizzrobes.  They're extremely quick and cast spells all over place.  But if you're short on bombs, killing Wizzrobes will replenish that supply rather quickly.

From the front room of the labyrinth, make a left turn and make sure the red candle is equipped.  The room that Link will enter is dark.  Best bets against Wizzrobes is to avoid them if at all possible.  Next, turn north, and keep going north until you eventually enter the room A3 on the grid.  

When you enter this room with the 5 Wizzrobes (bottom left), the shutterdoor that you entered through will close.  Going through the door at top will require you to face Gleelok, an encounter you must beat.  (If you are going swordless, fighting Gleelok is not an option since you can't hurt him because you don't have the only other weapon that can.)  The wall on the right side of this room is bare - try planting a bomb there.  A door will now appear.  Go through this door.  Use the stepladder to get the key that would unlock the door to the north.  

In that next room get the map (but do not make that left turn through that open door) and continue north one more room to get the Magic Wand, if you have an extra key.  If you don't, the room just east of this room does.  The next room should have Wizzrobes.  Eliminate all the Wizzrobes to open up the passage for the Magic Wand.  

Now backtrack to the room that you entered through a door that the bomb created (bottom right in the image above).  Now go south from that room.  The next room, you must defeat all the Vires to get the shutterdoors to open.  Now go east.  Wipe out the Wizzrobes and Like Likes from this room.  When they are gone, push the block to open the passage.

Now Link should be in the other side of Dragon.  Ignoring all the enemies, go south one room (making sure that you get the key), west one, and then go north through the locked door.  In that room, equip with the Bow/Arrow.  Now Link should be facing Gohma.  Gohma is very nasty, and not even the Magic Shield can block it's fireballs.  When Gohma's eye is open, give him one arrow to the eye.  Gohma should be eliminated if that is done correctly and the container heart will appear.  Take that, and continue north for the Triforce fragment.