Cooper wins Olympic freestyle moguls gold after dramatic tie-break
The athletes both scored 83.71 in their final runs in Livigno, meaning the marks they achieved for their turns came into play.
Skiers are judged on three criteria -- turns (making up 60 percent of the score), aerial tricks (20 percent) and speed (20 percent).
Woods' mark of 48.40 was marginally better than Kingsbury's 47.70, giving the 25-year-old a first Olympic medal.
Reigning world champion Ikuma Horishima of Japan won bronze with 83.44, performing a sensational 1440 on the final jump.
Just 0.27 points separated the top three in a scintillating end to the competition at the Milan-Cortina Games.
Woods was in first place out of 20 competitors after Final 1, meaning he was the last to go of the eight skiers in Final 2.
Kingsbury, the Olympic champion from Pyeongchang 2018, produced a storming run to usurp the high-flying Horishima and take top spot with just Cooper to go.
But the Australian kept his calm under intense pressure, screaming in delight when he realised he was Olympic champion at his second Winter Games.
Sweden's defending Olympic champion Walter Wallberg finished in fourth spot.
"It hasn't sunk in yet," said Cooper. "Speechless, super emotional, very proud.
"I didn't have any expectations this morning. I just wanted to go out and ski my runs and stay true to what I know I can achieve."
Despite the pain of missing out on gold, Kingsbury became the first athlete to win medals in the same freestyle skiing event at four straight Olympics.
In addition to his gold in 2018, he won silver at the 2014 Sochi Games and again eight years later in Beijing.
Kingsbury said it felt "amazing" to win a fourth Olympic medal.
"I'm very happy with my skiing," he said. "It was close, a tiebreak –- unfortunately I'm the guy not on the good side of it. But I've worked very hard for this medal.
"I'm getting older, I'm 33, I had an injury in September. At some point it felt like it was impossible to be back at that level."
In moguls, athletes navigate a steep course full of bumps, combining technical turns, jumps, aerial manoeuvres, and speed.