Australia scored 14 runs off the final over to win the first of two Twenty20s against India in a low scoring game.
India failed to build on a foundation set by KL Rahul, who scored 50 from 36 balls, as they stuttered to 126-7.
Australia looked to be in control before Glenn Maxwell was out for 56 and they lost five quick wickets to leave them 113-7 going into the last over.
But Jhye Richardson and Pat Cummins each scored seven from three balls off Umesh Yadav to help the visitors win.
India captain and star batsman Virat Kohli was caught at long-on for 24 in the ninth over to end a stand of 55, but Rahul hit six fours and a six before he spooned a catch to mid-off, one of two wickets to fall in the 13th over.
Australia lost two wickets in two balls early in their innings but Maxwell had a spell of four fours and a six in six deliveries to propel the run chase, only to hole out to long-off midway through the 14th over.
Richardson and Cummins both struck Yadav for boundaries in the last over, the latter hitting the ball back past the bowler to complete the winning two runs from the final delivery.
The second and final T20 is in Bangalore on Wednesday, with a five-match one-day international series beginning on 2 March.