Williamson (17) won the bronze medal in the 20km event, and hopes for another medal in the omnium event over the weekend.
Italian Maria Giulia Confalonieri won the race with 43 points after lapping the field about halfway through the race and Mexican Ingrid Drexel won the silver medal with 23 points. Williamson collected 15 points to finish third.
Sophie's father and co-coach, Martyn Williamson, said he had no idea how his daughter had gone in the race until she Skyped the family after the race.
"When we saw that medal around her neck and the smile, we knew she had done well," he said.
"She had the intention of going out early and trying to get as many points in the early sprints as possible. By the last sprint she was pretty much spent.
"Perhaps, if she looks back on the race, she might think she had gone out a bit too early and made herself vulnerable. But it is that sort of race where anything can happen. She was really excited to win a medal."
She won points in five of the 10 sprints within the race.
The event was on a 333m track, bigger than the usual 250m size track riders use, with steep banks.
Williamson has little time to rest on the success, as she lines up in the first three events of the omnium tonight. The six-race event is cycling's version of the decathlon and heptathlon, where a rider's all-round riding ability is tested.
As the omnium is an Olympic event and the women's points race is not, Williamson has been targeting the multi-race event.
After finishing at the championships in Moscow, the Dunstan High School pupil will spend some time with relatives in the United Kingdom and then head to Belgium to train with the New Zealand team.
Late next month, she will race in the world junior road championships in Copenhagen, Denmark. It may be something of a Williamson benefit, as her older brother James is a candidate to make the under-23 team after some good results in the United States.
