Hazlewood initially led the way at The Gabba, completing a five-wicket haul as the tourists stumbled from an imposing overnight score of 311-4 to 408 all out.
Australia then dashed to 221-4 in reply before bad light forced an early close, with Smith - skippering the side for the first time in the absence of the injured Michael Clarke - striking 65 not out.
Bat had dominated ball on a scorching opening day in Brisbane, which saw a number of Australia's pace bowlers hampered by injury or fatigue.
Hazlewood was among those to suffer discomfort, but bounced back on Thursday by striking on three occasions to finish with impressive figures of 5-68 from 23.2 overs.
The tall seamer struck in the third over of the morning with a wonderful delivery that moved away late and induced an edge behind from Ajinkya Rahane (81).
Only seven runs were added before Rohit Sharma (32) was superbly held at second slip by Smith off Shane Watson (1-39).
A partnership of 57 between returning skipper MS Dhoni (33) and Ravichandran Ashwin (35) steadied the ship for India, but Hazlewood was not finished.
The 23-year-old had Ashwin caught at first slip and another batsman held behind the wicket when Dhoni got an inside edge through to Haddin, providing the wicketkeeper with a sixth catch of the innings.
Nathan Lyon wrapped things up with the dismissals of Varun Aaron (4) and Umesh Yadav (9) in successive overs, the former falling to a superb catch at short-leg by substitute fielder Marnus Labuschagne.
After losing their last four wickets for 23, India saw Australia make a flying start with the bat, thanks largely to the typically positive David Warner (29).
However, Warner and Watson (25) both fell when apparently well set. The opener top-edged an attempted pull off Yadav (3-48) to be caught by Ashwin, who subsequently removed Watson courtesy of Shikhar Dhawan's catch at mid-on.
Chris Rogers brought up a welcome half-century, but was then caught down the leg side to become Yadav's second victim.
That left Australia 121-3, but Smith counter-attacked superbly, striking six fours and two maximums in a continuation of his recent fine form.
A fourth-wicket stand of 87 ended when Shaun Marsh was caught at first slip off Yadav, but Mitchell Marsh (seven not out) accompanied Smith through to the close, which arrived with Australia 187 in arrears.