After starting day three in Delhi with a 62-run lead with nine wickets in hand, Australia looked on track to build a matchwinning advantage in seeking to level the series at 1-1.
But Jadeja (7-42) and Ashwin (3-59) quickly went about destroying the Australian line-up, unleashing carnage similar to that seen in the first Test in Nagpur.
Australia went from 1-61 overnight to all-out for 113 in less than a session, setting India a target of just 115 to win.
After appearing to be managing the tricky conditions at 2-85, the tourists lost 8-28, presenting India with a golden chance to go 2-0 up in the four-Test series and retain the Border-Gavaskar trophy.
Australia lost 4-0 in 11 balls to crash from 3-95 to 7-95 in a devastating period before drinks on Sunday morning at Arun Jaitley Stadium.
The tourists even lost three wickets in as many balls - one to Ashwin and two to Jadeja - when Matt Renshaw, Peter Handscomb and captain Pat Cummins were unable to contain India's spinning brilliance.
Handscomb got an unplayable delivery from Jadeja, but Cummins' shot was a wild, Twenty20-like swing to a ball that smashed into his stumps.
Given his first opportunity to open in Test cricket after David Warner was subbed out with concussion on Saturday, Travis Head added just four to his overnight score before he was out to Ashwin on the last ball of the first over.
Marnus Labuschagne (35) and Steve Smith appeared steady at the crease for a short period, but once the latter fell Australia crumbled on a rapidly deteriorating pitch.
Chasing has proved difficult at Arun Jaitley Stadium over the years, with the highest-successful fourth-innings score being 5-276 - in 1987 and 2011.
India are unbeaten in Tests at their Delhi fortress for the past 36 years.
