The 16-year-old won the balance beam and finished second in the floor exercise. On Saturday, she won the uneven bars final.
"I knew I could do well but this came as a surprise," said Onyshko. "I still have a lot of work to do, but this really makes me more confident for sure."
In the beam final, Onyshko scored 14.075 points to clinch the gold, well ahead of runner-up Lorrane Oliveira Dos Santos of Brazil with 13.525 points.
In the floor final Eythora Thorsdottir of the Netherlands won gold with a score of 14.125, followed by Onyshko with 13.650 points.
Victoria Woo of La Prairie, Que., placed fifth with 13.225 points.