Campbell went into the clean and jerk section of the competition in fourth place but successive lifts of 156kg and 161kg sealed her historic place on the podium.
China's Li Wenwen twice broke the Olympic record as she took gold and Sarah Robles of the US had to settle for bronze.
Laurel Hubbard, the first openly transgender athlete to compete in the Olympic Games, was eliminated after the snatch section of the competition after failing to register a lift.
The 27-year-old Campbell took bronze at the 2018 Commonwealth Games but announced her move into medal contention when she won this year's European Championships in Moscow.
And she was able to head into the competition somewhat under the radar given the media circus surrounding Hubbard, whom Campbell always maintained fully deserved her place in the competition.
But while Hubbard came up short with her first three lifts, Campbell moved into contention with a lift of 122kg in snatch, placing her behind Li, Robles and South Korea's Seon Mi-Lee.
A fail by Lee on her final attempt gave Campbell her chance, which she seized by lifting 156. When Robles was denied on 157kg, Campbell improved her colour once more by lifting 161kg.
As well as being Britain's first female weightlifting medallist, Campbell is also the first Briton to win a medal in the sport since 1984.
Hubbard did not take questions in the mixed zone following her elimination, but she read a statement.
She said: "I know that from a sporting perspective I haven't really hit the standards that I put upon myself and perhaps the standards that my country has expected of me.
"But one of the things for which I am profoundly grateful is that the supporters in New Zealand have given me so much and have been beyond astonishing.
"I'd like to thank the New Zealand Olympic Committee, they have supported me through what have been quite difficult times.
"I know that my participation at these Games has not been entirely without controversy but they have been just so wonderful and I'm so grateful to them."