After India got away to a flying start, Green latched onto a Shubman Gill edge low to the ground at gully with his non-preferred left hand off the bowling of Scott Boland.
Gill remained at the wicket and the decision was reviewed to the third umpire, before Richard Kettleborough determined Green had his hand under the ball and gave it out.
It is the second time Green has taken a great catch in the match, after also claiming a screamer in the first innings to dismiss Ravindra Jadeja off Pat Cummins.
It came after Boland was handed the new ball ahead of Mitchell Starc, in the Victorian’s pursuit of selection for the first Ashes Test.
It marked only the second time Starc has not opened and instead bowled first-change for Australia since 2015.
Australia have given themselves 137 overs to bowl out India and claim the global title, after Alex Carey’s unbeaten 66 ensured Australia could post a large target.
No side in Test history has pulled off a run chase greater than 418 in a fourth innings, while the previous highest effort at The Oval was 263 way back in 1902.
If the match is drawn, both sides will share the trophy.
Supported by Green (25) and Starc (41), Carey put away the reverse sweep and batted patiently on a wicket with plenty of uneven bounce on Saturday.
After being dismissed on the shot four times this year against India, he did not play it once and instead nurdled Jadeja (3-58) around through his 105-ball knock.
His runs instead came through the offside, cover-driving Jadeja for one boundary and hitting Mohammed Siraj through the region on the front and back foot.
The South Australian will enter next week’s Ashes after his best Test match with the bat, having scored 48 in the first innings before being dismissed reverse sweeping.
Australia will also be happy with the time Green spent in the middle.
He soaked up 95 balls in his innings, readjusting to Test cricket mode after arriving in England following two months in the hurry-scurry of the Indian Premier League.
He was out in the first innings driving at a ball well away from his body, but was far more composed in the second.
With his feet moving well, he played the shot of the morning when he sent a Mohammed Shami ball back past the Indian quick for four.
But he eventually became tied in a web by Jadeja, bowled when a ball bounced off his gloves and onto the stumps after he attempted to shoulder arms to the spinner.
When he went Starc was typically more aggressive, with his 41 coming from 57 balls as he used his long levers to good effect against the new ball late in the innings.