Gu, the defending champion and one of Milano Cortina's highest profile athletes, had to settle for silver, while Italy's Flora Tabanelli took bronze.
Oldham, 24, came into the final with the wind in her sails, having topped the qualifiers, and fresh from a slopestyle bronze on a week ago.
"I'm on cloud nine right now. I'm still, I think, processing everything. I don't even think the emotion has fully hit me yet, but so happy, for sure," she said, dedicating the victory to her brother Cody, who turned 18 on Monday.
In November, Oldham suffered a concussion while practising new tricks in Switzerland, and was off skiing for six weeks, a period she previously described as "really tough".
Second silver for Gu
For American-born Gu it was the second silver medal of these Games, after runner-up spot in slopestyle. She has another chance of a gold in the halfpipe competition later this week.
Olympic novice Tabanelli, 18, was the youngest contender at the final, and the reigning freestyle world champion. Her bronze capped another successful recovery from injury, as she had torn a knee ligament in November.
She came to the Games with high hopes along with her brother Miro, who failed to make it into Tuesday's men's big air final.
Swiss contender in hospital
The three medallists hugged and smiled on the podium with their hardware, before Tabanelli took a group selfie, while the home crowd chanted "Flor-A, Flor-A".
In big air, skiers launch themselves off a jump and perform as many flips and twists as they can before landing, reaching speeds of up to 60 kilometres per hour. Judges assess height, technique, originality and other factors.
The start of Monday's final was delayed by over an hour due to a snow blizzard that would have made it unsafe to compete.
One of the top medal hopefuls, Mathilde Gremaud of Switzerland, pulled out just before the start due to an injury after a crash in a practice run.
Gremaud took a big hit on the hip and will remain in hospital under observation overnight a spokesperson for the Swiss team said.