Defending champion Hancock edged out compatriot Conner Prince, who settled for the silver, while Taiwan's Lee Meng Yuan took bronze.
Yang and Jedrzejewski were tied on 37 points at the top forcing the shoot-off in which the Korean prevailed 4-1 in her Olympic debut at the Chateauroux Shooting Centre.
"This gold medal has proved I am the world's best right now," said the bespectacled Yang.
"I will have to train harder to try to put the flag highest once again in LA (in 2028)."
Local favourite Jedrzejewski said she struggled to control her emotions in the final shots.
"I lost maybe because my emotions were running very high, but I'm happy with my silver," she said.
"There was a lot of emotion in the hall, and it was a really intense competition."
Hungary's Veronika Major won the bronze also via a shoot-off, denying India's Manu Bhaker a third Paris Games medal.
"I was very happy to go to the final in the first place," Major said.
"This is the greatest moment of my career so far."