The visitors stretched their second innings lead to 298 on a viciously turning track.
Touring side reach 256-9 at stumps on day one of the first Test at Pune.
India are sitting pretty at the top of the Test rankings after six consecutive series victories.
The leg-spinner captured three for 25 at a crucial stage as Australia went on to win by 41 runs.
The third match is in Wellington on Saturday.
The England all-rounder attracted multiple bids from the rival franchises competing in the eight-team tournament.
The 36-year-old star had already quit Tests in 2010 and ODI cricket after the 2015 World Cup but still skippered the Pakistan Twenty20 team at the 2016 world championships in India.
The dashing batsman pounded the Australian bowlers with a 46-ball 84 to get his side home by two wickets.
The South African captain scored 37 not out to earn his team a 1-0 lead in the five-match series.
The win was made possible by a dashing 52 off 37 balls by Asela Gunaratne, who was named man-of-the-match.
The leg-spinner takes career-best figures to condemn New Zealand to their first loss at home this summer.
The right-handed batsman scored 1,485 runs at an average of 61.87, ranking him second on the all time list behind Bradman's incredible 99.94.
Durham's 25-year-old all-rounder Ben Stokes was named vice-captain, the role previously held by Root.