Playing in his second Test, Kumar (three for 36) removed both openers and Shane Watson as Australia, 1-0 down in the four-match series, once again left captain Michael Clarke (20no) with a mountain to climb.
Australia made two changes from the side which lost the opening match in Chennai with Glenn Maxwell making his Test debut for Mitchell Starc, while Xavier Doherty replaced Nathan Lyon. The home side were unchanged.
Kumar bowled David Warner with a delivery which cut back into the left-hander, then trapped Ed Cowan lbw with a ball which may have pitched outside leg stump.
That left Australia 15 for two, with Phillip Hughes and Watson at the crease.
The pair effected a mini-recovery, with the left hander playing some eye-catching late drives off the back foot, but Watson (23) then fell lbw attempting to to a ball which kept a little low with the score on 57.
Hughes followed, caught behind at the second attempt by Mahendra Singh Dhoni off the bowling of Ravichandran Ashwin, leaving the defiant Clarke, who struck Ashwin for a straight six shortly before lunch, alongside Matthew Wade to pick up the pieces.