Canada controlled much of the possession before Landry, from Toronto, raced the length of the pitch to give Canada the victory. It was the fourth straight time that Canada has finished third at a World Rugby Women's Sevens Series event.
Kayla Moleschi of Williams Lake, B.C., scored the other Canadian try late in the first half.
New Zealand shaded Australia 19-17 in the championship game.
Earlier in the day, Canada whipped the United States 36-0 in the Cup quarter-finals before dropping a 29-7 decision to Australia in the Cup semifinals.
Canadian head coach John Tait said he's looking for a more consistent effort going forward.
"We now need to become clinical against the top sides and not panic and try to recover a score by forcing play we're not fully prepared to execute on," said Tait. "The onus is on me and the rest of the leadership group to develop ways to get us back on plan when this happens. The teams in the series are much improved and we need everyone to bring their individual excellence into a cohesive attack and defence to get those semifinals and finals wins."
The result gives Canada 16 points in the standings and they will now be a top seed at the Brazil Sevens in February. The top four teams at the end of the season will automatically qualify for the 2016 Summer Olympics.
In men's play, World Series leader Fiji and defending champion New Zealand reached the quarter-finals with unbeaten records Friday while Canada did not qualify for Cup competition.
South Africa and Australia swept their pools to advance to the Cup quarter-finals on Saturday. It will be Fiji vs. Wales, Australia vs. Scotland, South Africa vs. Argentina, and New Zealand vs. England, a rematch from the series opener at the Gold Coast in October, when England thrashed the Kiwis.
Canada started the day by upsetting Wales 19-14 before dropping a 24-12 decision to South Africa and then falling 26-7 to Portugal.
Canada will settle for an appearance in Saturday's Bowl quarter-final against France.
Fiji, winner of the Gold Coast title, dispatched France and Brazil, and was pressed by Argentina, which had already qualified.
Fernando Luna exploited a Fiji error in front of its posts, scoring a converted try for Argentina to lead 21-19 with four minutes to go. But that was plenty of time, as Semi Kunatani made a big fend en route to a try to regain the lead, and Fiji prevailed 33-21.
New Zealand didn't drop a point while also scoring a combined 120 against Japan, Scotland and Samoa.
Samoa, the runner-up on the Gold Coast, needed to beat New Zealand for a chance to advance on points difference, but was already three tries down by halftime and lost 41-0. Scotland, which lost to Samoa, advanced instead.
South Africa defeated Portugal, Canada, and Wales, the latter by a score of 12-5.
Wales received a lifeline when Portugal beat Canada. That left Wales going into its pool finale knowing it could advance with a loss to South Africa by fewer than 21 points. Wales had beaten South Africa only once.
South Africa came from behind, but Wales threatened at least a draw in the dying stages. South Africa's defended superbly, especially a Cecil Afrika tackle on Craig Price, who couldn't offload to a teammate outside him.
Australia conceded an early Dan Norton try, but tied England by halftime, and won 12-5 with the only try of the second half to win its pool and avoid a resurgent New Zealand.
England, meanwhile, face the Kiwis with injury worries, after Dan Bibby joined James Rodwell on the sidelines with an apparent shoulder complaint.
With files from The Associated Press.