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The defending champions scored five tries to move on to the quarter-final where they will meet France.
Joseva Ravouvou was the standout man for New Zealand, chipping in with two tries in a dominant display of power and pace.
Russia managed to score a late try in the dying stages of the first half to stay within seven points going into the break, but weren't able to keep up in the second period.
They will face a tricky France side who upset Australia 22-17 in a thrilling round of 16 encounter.
Earlier in the day, the Black Ferns Sevens eased into the World Cup semifinals with two comfortable wins on day one of the event.
Under the new knock-out format there was never any doubt that the defending champions would be eliminated as they opened the day with a 57-0 thrashing of Mexico before an equally dominant display in the quarter-finals, beating Ireland 45-0.
New Zealand ran in 16 tries in the two games with Michaela Blyde managing a double and a hat-trick across the fixtures.
New Zealand captain Sarah Goss scored a hat-trick in the first game as Mexico struggled to keep up with the Olympic silver medallists and defending World Cup winners.
Blyde and Portia Woodman both scored doubles while Kelly Brazier and Stacey Waaka also crossed over as Tyler Nathan-Wong added six conversions.
Mexico failed to carry the ball over the halfway line and New Zealand created plenty of space out wide.
It was much of the same in the quarter-final against Ireland, who earned their spot in the final eight with an upset win over England.
Blyde scored a hat-trick while Woodman once again picked up a double. Niall Williams was impressive setting up a Blyde try before scoring one of her own. Gayle Broughton then came off the bench and twice put Blyde away to the tryline.
The Black Ferns Sevens will play USA in tomorrow morning's semifinal.