Having finally bowled their hosts out for 572 on the fourth morning, Michael Clarke's men set about chasing down a 158-run deficit but were staring at an innings defeat as Ashwin steadily made his way through the batting line-up.
However, a half century from Moises Henriques (75 not out) and an unbeaten last-wicket stand of 57 with Nathan Lyon (eight) saw Australia past their target and left India needing to bat again to wrap up the win.
At close of play on day four, the tourists were on 232 for nine, a lead of 40 runs, with Henriques not out on 75.
India resumed on 515 for eight and added 57 more in the morning before being bowled out for 572 – a daunting lead of 192.
MS Dhoni, the scorer of an unforgettable double century on Sunday, was finally out for 224 when he nicked James Pattinson to Matthew Wade at the wicket.
But if Australia thought their work in the field was done, India had other ideas as Bhuvneshwar Kumar and Ishant Sharma added 26 more for the last wicket.
Debutant Bhuvneshwar finished with 38 before Peter Siddle wrapped things up.
Pattinson's five for 96 was by some distance the best return on an otherwise messy bowling card.
Australia faced a tough challenge on a pitch that had started to turn and were given a further obstacle when David Warner was deemed too ill to open.
Shane Watson replaced him in his old position and he, along with Ed Cowan, was confronted by spin at both ends from the off.
* Press Association