Australia wobble in pursuit of World Cup final victory
Jasprit Bumrah deceived Smith with a slower ball that had the right-hander lbw, the India paceman celebrating wildly with his team-mates as a near full house of around 130,000 fans erupted with joy.
Smith decided against a review and his exit left Australia 47-3 from seven overs, chasing a target of 241 to win a record-extending sixth one-day international World Cup title.
Australia were 60-3 at the 10-over mark, with opener Travis Head 19 not out and Marnus Labuschagne yet to get off the mark.
Earlier, India, bidding for a third World Cup title and second on home soil after their 2011 triumph, were dismissed for 240 after being sent into bat.
Left-arm quick Mitchell Starc took 3-55 and Australia captain Pat Cummins a miserly 2-34 in his maximum 10 overs.
KL Rahul top-scored with 66 but hit only one boundary in 107 balls.
Key batsman Virat Kohli made 54 after India captain Rohit Sharma struck a fluent 47 off just 31 balls opening the innings.