When you enter the chamber and the four flames descend, let them come all the way down. Then, run straight up as far as you can go. When the flames are almost touching you, run left all the way, and again wait until the flames almost touch you (otherwise, they'll loop around and corner you). Loop around clockwise until you reach the intersection, then head right and then down. When you can go right again, go right and then turn up at the intersection. The key is to trap the first four flames in the left side of the maze so they can't chase you up into the final section.
When you reach the next intersection, go up all the way, and wait for the flames to the left to form a perfect line. Then go back down to the intersection, and again, wait until the flames are almost on you (at least one of them might still be stuck in the "pocket" to the left). This time, run around the loop counterclockwise, always making sure the flames are close behind, or they'll break away and corner you. As you run around the loop, any flames still trapped in the pocket to the left will break out and join the line of fire trailing you. Keep running counterclockwise until you've got a clear shot to enter that "pocket" to the left, and once you can go there, follow the path up to the exit.