Twenty-three-year-old Campbell sat in third place after succeeding with a 103kg snatch attempt, but her clean and jerk best of 139kg was not enough to prevent Amoe-Tarrant from usurping her of second place.
Amoe-Tarrant produced a clean and jerk of 142kg - the best in the competition - to claim silver, while Samoa's Feagaiga Stowers stormed to gold.
Stowers produced a 113kg snatch and 140kg clean and jerk to end with a combined total of 253kg - 10kg clear of second place.
The Samoan was aided by the withdrawal of favourite Laurel Hubbard, though, with New Zealand's first transgender competitor being forced to withdraw following a nasty-looking injury on her final attempt at the snatch.
Hubbard's first successful snatch attempt of 120kg was comfortably the best in the competition, but she tried to push for an even bigger lead and paid the price, failing with a 127kg effort before sustaining the injury on her 132kg attempt.