Zavarzina made her World Cup debut on 23 October 2006 in a parallel slalom race in Landgraaf where she finished in 31st place and scored her first World Cup points. It was her only World Cup start of the season and she mainly participated in Europa Cup for the next couple of seasons. She won three Europa Cup races during the 2007/08 season and placed third in the overall rankings. She won a bronze medal in parallel giant slalom at the 2008 Junior World Championships in Valmalenco. In the following season, despite managing only one victory in Europa Cup, Zavarzina reached the podium five times and finished the season as the runner-up in the overall standings. With that promising performances in the Europa Cup, Zavarzina was given a spot in the parallel giant slalom race at the 2009 World Championships in Gangwon, but she was unable to qualify for the elimination round after finishing 21st in qualification round. At the 2009 Junior World Championships in Nagano, she won a silver medal in parallel slalom, the discipline that she made only one podium in her Europa Cup career. She made four appearances in the World Cup during the 2008/09 season but passed the qualification round only once, in Landgraaf where she made her World Cup debut.