Sweden's Emma Dahlstom, who qualified in first place with a 91.40 - the highest score of the day - also fell on the first kicker.
Switzerland's Sarah Hoefflin won gold with a final run that included one of the first switch double-cork tricks every thrown by a woman in slopestyle competition.
Atkin was awarded 79.40 for her second run to move to third entering the final run, which she began knowing she needed to improve to get on the podium.
In doing so, Atkin (84.80 pts) becomes the first British freestyle skier to stand on the Olympic podium.
"I'm just really happy I can be representing Team GB", she said, draped in a Union Jack.
But Tsubota crashed on her second run and couldn't take advantage of her third, eventually falling out of medal contention.
"I was so stoked, that was really incredible", said Gremaud.
Indeed it was team-mate Katie Summerhayes who was showing the greater promise with a less hard but cleaner first run down the course.
Gremaud outlined her status as a future star of the sport as the 18-year-old's total was enough for silver.
"It is not the result that I wanted but it is something I am really grateful for", said Howell, adding that the speed on the course had surprised her. "That helped me a lot".
The bronze medallist from Sochi, Quebec City's Kim Lamarre, also failed to advance to the final. Like Jenny Jones on her snowboard four years ago, Atkin reined in her natural instinct to go for broke in favour of a less spectacular, but neat and clean final run of the women's ski slopestyle and, like Jones, she reaped the reward for her composure.
Although she hasn't declared a major yet, Atkin is leaning toward the "idea of either math or physics, but I don't want to get too specific yet - mainly because the liberal arts on the Block Plan encourages me to experiment and explore as I go".
Dahlstrom failed to land on the first jump however - the Swede crashing to the ground - leading to celebrations from a large Team GB contingent at the foot of the course.
"I went up for the finals and made a decision to push it a bit more because I wanted to do myself justice".