Australia went to lunch on day four at 2-53, leading India by 158.
Veteran opener Usman Khawaja avoided falling victim to Bumrah for the sixth time in the series.
But when Khawaja (21) was looking set, the left-hander was bowled by Mohammed Siraj, who shushed the crowd, with Australian fans booing him since he gave Travis Head a send-off in Adelaide.
Mohammed Siraj gestures to the crowd after dismissing Usman Khawaja. (James Ross/AAP PHOTOS)
Marnus Labuschagne (20no) and Steve Smith (2no) will resume batting after lunch.
After writing himself into Boxing Day folklore with an audacious 60 on Thursday that included a range of T20-style ramp shots, Konstas was out for eight in his second Test innings when Bumrah rattled his stumps.
The 19-year-old has enjoyed every minute of his Test debut, interacting with the crowd, and taking selfies with spectators.
When Bumrah dismissed Konstas, the world's No.1 Test bowler threw his hands up in the air to the strong Indian contingent sitting in the Ponsford Stand.
The 31-year-old's celebration seemed to mimic what Konstas had been doing with the crowd while fielding.
Australia started their second innings with a lead of 105 after bowling India out for 369 just 17 minutes into play on Sunday.
A brilliant maiden century from Nitish Kumar Reddy powered India past the follow-on target on Saturday, after the tourists were in trouble at 5-164 in reply to Australia's 474.
Nathan Lyon (3-96) removed Reddy (114) when the breakout star hit an easy catch to long-off.
Nathan Lyon removed century-maker Nitish Kumar Reddy, finishing the innings with 3-96. (James Ross/AAP PHOTOS)
It came after a controversial call from the TV umpire in the previous over when No.11 Siraj was given not out despite the ball appearing to carry to slips.
Australia captain Pat Cummins wanted to review the decision but was unable to because TV umpire Sharfuddoula was the one who made the call.
Reddy, just 21 and playing in his fourth Test, helped India add another 11 runs to their overnight score of 9-358.
Australia made all the early running in this fourth Test, but India turned it around on Saturday through Reddy and Washington Sundar (50) to have their best day since dominating the series-opener in Perth.
Scott Boland, the pick of Australia's bowlers with 3-57 from 27 overs, said team morale remained high.
Australia bowled 120 overs at India as the effects of a gruelling series appear to take their toll on Mitchell Starc.
The star quick received treatment on his back on day three, but continued to bowl at top pace for Saturday's final session.
Starc has toiled for 25 overs in this Test, backing up from the 24 he bowled in Brisbane.
Mitchell Starc appeared to show the effects of a tough campaign on Saturday. (James Ross/AAP PHOTOS)
Boland was recalled to the XI after star quick Josh Hazlewood broke down for a second time in the series at the Gabba.
The fitness of Australia's bowling attack will be a major factor in deciding the outcome of the Border-Gavaskar trophy.
This Test is set to go five days, a rarity in modern cricket in Australia, meaning the players will have only three days off before the finale at the SCG gets underway.
With the five-match series tied 1-1, India will retain the Border-Gavaskar trophy if they pull off an extraordinary comeback victory in this Test.
Australia need to prevent the tourists from winning one of the last two matches of the series, while claiming at least one themselves, to regain the trophy for the first time since 2014-15.
Reserve batter Josh Inglis has been ruled out for the rest of the series after suffering a calf strain while fielding on day two.
Inglis' injury is a major blow for the Perth Scorchers and their bid for BBL success this season.
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