The 22-year-old Olympian saw off China's Yang Zhou, who was later disqualified, for a podium place, capping a busy week in Debrecen, Hungary.
Christie also finished fifth in the 3,000m superfinal after crashing out in the semi-final stage of the 500m on Saturday.
The Scot, Team GB's first-ever World Cup champion, finished the 1,000m final in a time of 1:32.281 minutes – while China's Meng Wang took gold in 1:31.549.
Silver went to Holland's Jorien ter Mors who also claimed bronze in the 3,000m superfinal.
In the men's 1,000m final Da Woon Sin claimed gold while in the in the 3,000m superfinal he was forced to settle for silver by South Korean team-mate Yun-Jae Kim.