Bumrah scythed through the middle order on day four in Melbourne to record figures of 4-56 off 24 overs and hand India an outside chance of victory in a match the home side have mostly dominated.
However, India's hopes were diminished by a stubborn, unbeaten final-wicket stand of 55 from 17.5 overs.
Nathan Lyon was at the crease on 41 while number 11 Scott Boland had seen off 65 balls to be 10 not out at the close.
Up by 105 runs on the first innings, Australia were rocked by losing four wickets for 11 midway through the second session to slump to 91-6.
Marnus Labuschagne and Pat Cummins responded with a vital partnership of 57, taking the steam out of Bumrah's assault.
A charmed Labuschagne scored 70, to back up his first-innings 72, while Cummins produced a captain's knock of 41 -- giving him a career-high combined 90 runs for the Test.
Both teams are seeking to go 2-1 up in the series heading into the fifth and final Test in Sydney.
Australia remain in the stronger position on a pitch showing signs of variable bounce.
"Their middle to lower order batting has been exceptional for the last four or five series we've played against them, so never comfortable with any amount of runs," said Labuschagne after play ended.
"At one stage we probably would have taken a 270 lead, and now the lower order has just put us in an amazing position with 333.
Brilliant Bumrah
The conditions were exploited brilliantly by Bumrah, who continued his dominance across a series in which he has claimed 29 wickets.
The 31-year-old powered past 200 Test victims and he boasts the lowest average at 19.45 of all 85 bowlers to have reached that milestone.
Labuschagne was beaten on numerous occasions by Bumrah's jagging deliveries but kept his nerve in a 139-ball knock.
The Australian number three was fortunate, on 46, to survive a straightforward dropped chance in the slips by Yashasvi Jaiswal.
Jaiswal also couldn't grasp a sharp chance under the helmet offered by Cummins in the final over before tea.
Labuschagne was trapped lbw by lively seamer Mohammed Siraj before Mitchell Starc was run out for five.
Spinner Ravindra Jadeja had Cummins caught behind, heralding the 10th-wicket stand which could yet prove pivotal to the outcome.
Siraj sparks collapse
Siraj, who took 3-66, sparked the earlier collapse, dismissing first innings centurion Steve Smith for 13.
Bumrah was at his best when removing Travis Head, Mitchell Marsh and Alex Carey cheaply in rapid-fire fashion.
All-rounder Marsh was caught behind without scoring, continuing a series in which he has made little meaningful contribution.
"It just happened so quickly," said Labuschagne of Australia's middle-order collapse.
"As a new batter coming in, it's very difficult to start in any Test match, but let alone when Bumrah's got his tail up with the ball nipping around."
Australia lost both openers before going to lunch at 53-2.
Teenage opener Sam Konstas was unable to match the fearless deeds from his debut first innings of 60, this time bowled for eight by Bumrah.
Usman Khawaja was bowled for 21 by Siraj, having been dropped early on by the butter-fingered Jaiswal at leg gully.
India were earlier dismissed for 369 in reply to Australia's first innings 474, after adding 11 to their overnight score, with maiden centurion Nitish Kumar Reddy the last man out, caught in the deep for 114.
Reddy, playing just his fourth Test, finished with the highest Test score in Melbourne for any player batting at number eight or lower.
He believed India could successfully chase a "big total" if they apply themselves.
"If you ask me, everyone is capable of playing on this pitch," he said.
