The Russian fell apart in the second set after Korda won the first on a tiebreaker taken to the brink, with the match finishing 7-6(8), 6-2.
The world number three fell three games behind in the second set, but looked like he might be about to make a comeback with a blistering win in the fourth game.
But the American Korda fought back, taking the next game, and Medvedev grew increasingly agitated, gesticulating in frustration and smashing his racket.
"I feel awesome, I played an incredible match," said Korda, thanking the crowd for their enthusiasm.
The match's thrilling rallies drew gasps and exclamations from spectators as the two battled it out.
"We played a really good tiebreaker, you know, it was up and down from the beginning... I'm happy to be on the winning side of it!" Korda added.
Medvedev had said the game would be "interesting" for him beforehand, referencing the fact that Korda beat him at their last meeting.
On Saturday, he also said he thought either "the court or the balls were slow", but that he wasn't sure which one.
Medvedev's elimination was watched from the courtside by tournament top seed Carlos Alcaraz, who is not playing Sunday.
The other main contender for the title, China Open champion Jannik Sinner, takes on Argentina's Sebastian Baez in the evening session.
Ruud through
World number nine Casper Ruud continued his smooth progress through the tournament with another straight-sets win, this time against the United States' Christopher Eubanks.
Ruud appeared in control throughout against the 32nd-ranked American, with the third-round match ending 6-4, 6-2.
The Norwegian said he was pleased he'd held his nerve against a "tough, fast" player.
Ruud said he was looking ahead to qualification for the ATP Finals in Turin, which takes place in November and features the world's best eight qualified players.
"I know every week is going to matter, and that's motivating and a bit stressful at the same time, but I like the battle and the competition," he said.
World number seven Andrey Rublev is through to the third round after defeating France's Quentin Halys.
Two top-20 players exited the tournament in the second round Sunday.
Britain's 16th-ranked Cameron Norrie was defeated by the United States' JJ Wolf, ranked 51st.
And 59th-ranked Lorenzo Sonego took out Frances Tiafoe, the world number 13, 2-6, 6-2, 6-3.
