They use a 32768Hz crystal like other watches. At least all modern watches do so.
The trick is in the divider, resp. the stopwatch counter: the 1/100 seconds are NOT exact, some are a bit too short, the rest too long, so that the total over 100 counts is exactly 1 second.
E.g. the EPSON S1C microcontroller has a special stopwatch timer peripheral exactly for this purpose.
This timer takes a 256Hz (=32768/128) input clock and divides it either by 2 or 3 to generate approximately 10ms intervals, so the 10ms intervals are exactly either 7.8125ms or 11.71875ms.
The divider pattern for 10 successive intervals (to get 1/10sec) is either:
a) 3-2-3-2-3-2-3-2-3-2 (=25/256sec=97.65625ms), or
b) 3-3-3-2-3-2-3-2-3-2 (=26/256sec=101.5625ms),
that is, the 1/10sec (100ms) also aren't exact.
The counter mixes these two patterns using a "b-b-a-a-b-b-a-a-b-b" scheme, resulting in a total of 256/256sec.
I'm sure all stopwatches use a similar scheme to generate the 1/100s intervals, maybe with a higher frequency basis clock, resulting in smaller deviations, but the principle is always the same.
Below is a copy of the respective page in the EPSON S1C Technical Manual that makes it a bit clearer (maybe...):
Here's a patent that describes in more detail how a 1/10s counter can be implemented:
http://www.freepatentsonline.com/3975898.html