The pair's emotional victory also marked the first female diving medal for Great Britain in 64 years.
Facing a challenging competition, Harper and Mew Jensen were in sixth place with only two dives remaining, yet a stunning final effort propelled them into fourth before a dramatic turn of events secured their bronze.
Australia, initially in contention for the bronze, made a critical error on their final dive, allowing the British duo to claim the coveted medal.
Mew Jensen, competing in her second Olympics, overcame significant adversity to achieve a remarkable result.
Just three months ago, she suffered a partial back fracture, leaving her with only a month to prepare for the Games.
"A month ago, I didn't think I would be here," a tearful Mew Jensen, 22, told the BBC.
"To be up on the boards and to come away with that, I can't imagine anything better."
Harper, celebrating her 24th birthday on Sunday, expressed her delight: "It is all we have worked towards.
"To come away with the medal feels really, really amazing."
China's Chang Yani and Chen Yiwen dominated the competition, securing their country's sixth consecutive gold medal in the event.
The United States' Sarah Bacon and Kassidy Cook claimed the silver.
