How do Zener diodes survive the breakdown during reverse bias?

The key difference between a Zener diode and a normal diode is that the Zener diode has a low breakdown voltage - typically in the few volts range. The breakdown voltage is low because the heavy doping means the depletion layer is very thin, and even at a low voltage the field strength over this thin depletion layer is very high. With a conventional diode the depletion layer is thicker and therefore a higher voltage is needed to achieve breakdown. However the breakdown phenomenon in a normal diode is basically the same as in a Zener diode.

Both diodes will be damaged if excessive current flows when the breakdown voltage is exceeded, and you need some resistance in series with the diode to limit the current.